Stand alone tasks are organized to support learning of content standards. These tasks can be used as initial instruction or to support students who are struggling with a particular topic.

- No Instructional Tasks at this time. If you have a suggestion, please contact the elementary mathematics specialist.

- Computations with Complex Numbers

This task asks students to perform computations involving complex numbers.

- A Lifetime of Savings

The purpose of this instructional task is to give students an opportunity to construct and find the value of a geometric series (A-SSE.4) in a financial literacy context. The task assumes that students have already developed the formula for a geometric series themselves; having them recognize the need for this formula (and look up if necessary) allows them to engage in MP 5, Use appropriate tools strategically. The task also provides students with an opportunity to look for and express regularity in repeated reasoning (MP 8), as the solution shows. This task also asks students to interpret the variables in the future value formula in the context of the problem (A-SSE.1). - Cantor Set

The purpose of this task is to use finite geometric series to investigate an amazing mathematical object that might inspire students' curiosity. The Cantor Set is an example of a fractal. - Course of Antibiotics

This task presents a real world application of finite geometric series. The context can lead into several interesting follow-up questions and projects. Many drugs only become effective after the amount in the body builds up to a certain level. This can be modeled very well with geometric series. - Triangle Series

The purpose of this task is to emphasize the adjective "geometric" in the "geometric" series, namely, that the algebraic notion of a common ratio between terms corresponds to the geometric notion of a repeated similarity transformation. - YouTube Explosion

The purpose of this task is to have students derive the formula for the sum of a specific finite geometric series. In determining the total number of views the YouTube video has, students will first come across the sum of the terms of a geometric sequence. Without a formula, students will have to calculate this sum by adding each term individually. By the end of the task, the students will have come up with a formula that will help them find the sum much quicker than by rote calculation.

- Non-Negative Polynomials

The task helps foster student understanding of the analogy described in the standard -- "Understand that polynomials form a system analogous to the integers..." -- in addition to having the same arithmetic operations available, there are many other instances in which integers and polynomials share common properties. - Powers of 11

This task might be used as either practice with polynomial arithmetic or as an introduction to the binomial theorem, providing a process for raising binomials to powers without dredging through many repetitive applications of the distributive law.

- Graphing from Factors I

The purpose of this task is to help students understand the relationship between the factors of a polynomial and the x-intercepts of the graph of the polynomial. By giving students two different polynomials with the same factors the task draws attention to the fact that both polynomials cross the x-axis at the same points. Students are then invited to reflect on why this is so by looking at the structure of the polynomials. - Graphing from Factors II

The purpose of this task is to give students an opportunity to see and use the structure of the factored form of a polynomial (MP7). - Graphing from Factors III

The task has students use the remainder theorem to deduce a linear factor of a cubic polynomial, and then to completely factor the polynomial. Students will need some procedure (e.g., synthetic or long division, or guess-and-check the coefficients) for determining the quadratic factor. Having the factored form permits students to deduce much about the structure of the graph. - Solving a Simple Cubic Equation

The purpose of this task is twofold. First, it prompts students to notice and explain a connection between the factored form of a polynomial and the location of its zeroes when graphed. Second, it highlights a complication that results from a seemingly innocent move that students might be tempted to make: "dividing both sides by x." - The Missing Coefficient

The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials. - Zeroes and factorization of a general polynomial

This task builds on ''Zeroes and factorization of a quadratic function'' parts I and II. The teacher may wish to recall the result from the first of these tasks, generalized to the polynomials of degree d considered here. - Zeroes and factorization of a non polynomial function

The level of the task is appropriate for assessment but since its intention is to provide extra depth to the standard A-APR.2 it is principally designed for instructional purposes only. - Zeroes and factorization of a quadratic polynomial I

For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by xr. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. - Zeroes and factorization of a quadratic polynomial II

This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions.

- Powers of 11

This task might be used as either practice with polynomial arithmetic or as an introduction to the binomial theorem, providing a process for raising binomials to powers without dredging through many repetitive applications of the distributive law. - Trina's Triangles

This task is a fleshing-out of the example suggested in A-APR.4 of the Common Core document, using the polynomial identity (x2+y2)2=(x2y2)2+(2xy)2 to generate Pythagorean triples.

- Combined Fuel Efficiency

The primary purpose of this problem is to rewrite simple rational expressions in different forms to exhibit different aspects of the expression, in the context of a relevant real-world context (the fuel efficiency of of a car). - Egyptian Fractions II

The purpose of this task is for students rewrite a simple rational expression and study the arithmetic of these expressions. Egyptian fractions provide an interesting context, both historically and mathematically, for students to use rational expressions.

- Basketball

This task provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable. - Bernardo and Sylvia Play a Game

This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game. - Buying a Car

The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types. - Clea on an Escalator

This task has students create equations to model a physical scenario, and then reason with those equations to come up with a solution. - Dimes and Quarters

This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. - Equations and Formulas

This task asks students to use inverse operations to solve the equations for the unknown variable, or for the designated variable if there is more than one. - Fishing Adventures 3

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically. - Global Positioning System I

This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres. - Growing coffee

This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. - How Much Folate?

This task a could be used as an introduction to writing and graphing linear inequalities. Part (a) includes significant scaffolding to support the introduction of the ideas. Part (b) demonstrates that, in some situations, writing down all possible combinations is not feasible. - Introduction to Polynomials - College Fund

This task could serve as an introduction to polynomials or as an application after students are familiar with this type of function. - Optimization Problems: Boomerangs

This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation. - Paper Folding

This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth. - Paying the rent

This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations. - Planes and wheat

This is a simple exercise in creating equations from a situation with many variables. By giving three different scenarios, the problem requires students to keep going back to the definitions of the variables, thus emphasizing the importance of defining variables when you write an equation. - Regular Tessellations of the plane

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible. - Rewriting equations

The goal of this task is to manipulate equations in order to solve for a specified variable. - Silver Rectangle

This task provides a geometric context for working with ratios and algebraic equations. Students will create and then solve an algebraic equation describing a remarkable shape, the silver rectangle. - Sum of angles in a polygon

This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180. - Throwing a Ball

Although this task is quite straightforward, it has a couple of aspects designed to encourage students to attend to the structure of the equation and the meaning of the variables in it. - Uranium 238

The goal of this task is to represent an exponential relationship by an equation and identify, using knowledge of the context and the structure of the equation, possible graphs for the equation. - Writing constraints

The purpose of this task is to give students practice writing a constraint equation for a given context.

- An Extraneous Solution

The goal of this task is to examine how extraneous solutions can arise when solving rational equations. - Basketball

This task provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable. - Canoe Trip

The goal of this task is to set up and solve an equation involving a simple rational expression. - Radical Equations

The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one. - Who wins the Race?

The goal of this task is to solve an equation involving a radical and then verify whether the solutions of the resulting quadratic equation are relevant.

- Introduction to Polynomials - College Fund

This task could serve as an introduction to polynomials or as an application after students are familiar with this type of function. - Optimization Problems: Boomerangs

This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.

- Throwing Baseballs

This task could be used for assessment or for practice. It allows the students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, we are asking the students to determine which function has the greatest maximum and the greatest non-negative root. - Which Function?

The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.

- Exponentials and Logarithms II

This task and its companion, F-BF Exponentials and Logarithms I, is designed to help students gain facility with properties of exponential and logarithm functions resulting from the fact that they are inverses. - Invertible or Not?

This task illustrates several components of standard F-BF.B.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which one is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not. - Latitude

This task requires students to use data to generate understanding of an invertible function. Some brief notes: First, the table has data ordered by percentage, not latitude, so students will have to reorder the data in order to generate a graph of N(â). Second, students are asked to interpret statements about inverse functions, for which an understanding of the quantities' units is particularly helpful. - Parabolas and Inverse Functions

This task assumes students have an understanding of the relationship between functions and equations. Using this knowledge, the students are prompted to try to solve equations in order to find the inverse of a function given in equation form: when no such solution is possible, this means that the function does not have an inverse. - Rainfall

In this task students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible. - Temperature Conversions

Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions). - Temperatures in degrees Fahrenheit and Celsius

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation. - US Households

The purpose of this task is to construct and use inverse functions to model a a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.

- Accuracy of Carbon 14 Dating II

This Illustrative Mathematics task is a refinement of "Carbon 14 dating" which focuses on accuracy. While the mathematical part of this task is suitable for assessment, the context makes it more appropriate for instructional purposes. This type of question is very important in science and it also provides an opportunity to study the very subtle question of how errors behave when applying a function: in some cases the errors can be magnified while in others they are lessened. - Bacteria Populations

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal. - Carbon 14 dating

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. - Carbon 14 dating in practice II

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies. - Exponential Kiss

The purpose of this task is twofold: first using technology to study the behavior of some exponential and logarithmic graphs and secondly to manipulate some explicit logarithmic and exponential expressions. - Graphene

This task provides a real world context for examining the incredible power of exponential growth/decay. - Newton's Law of Cooling

The coffee cooling experiment in this task is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations. - Snail Invasion

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year.

- Bicycle Wheel

The purpose of this task is to introduce radian measure for angles in a situation where it arises naturally. - Coordinates of Points on a Circle

The purpose of this task it to use geometry and algebra in order to understand the behavior of the trigonometric function f(x)=sinx+cosx. The task has been stated in an open ended fashion as there are natural solutions using geometry, or using the trigonometric identity sin2x=2sinxcosx, or algebraically solving a system of equations. - Equilateral triangles and trigonometric functions

The purpose of this task is to apply knowledge about triangles to calculate the sine and cosine of 30 and 60 degrees. - Exploring Sinusoidal Functions

This task serves as an introduction to the family of sinusoidal functions. It uses a desmos applet to let students explore the effect of changing the parameters in y=Asin(B(xâh))+k on the graph of the function. - Foxes and Rabbits 2

The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation. - Foxes and Rabbits 3

The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data. - Properties of Trigonometric Functions

The goal of this task is to use the unit circle and rigid transformations in order to establish some fundamental trigonometric function identities. - Special Triangles 1

Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angles, in this case of Ï/4. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss. - Special Triangles 2

Using known facts about the unit circle and isosceles triangles together with the Pythagorean Theorem, we can derive the sine and cosine of special angels, in this case of Ï/6. This task can be done as a mini lecture soliciting responses from the students, or as a challenge problem for students to ponder and discuss. - Trig Functions and the Unit Circle

The purpose of this task is to help students make the connection between the graphs of sint and cost and the x and y coordinates of points moving around the unit circle. Students have to match coordinates of points on the graph with coordinates and angles in the diagram of the unit circle. - Trigonometric Identities and Rigid Motions

The purpose of this task is to apply translations and reflections to the graphs of the equations f(x)=cosx and g(x)=sinx in order to derive some trigonometric identities. - Trigonometric functions for arbitrary angles

The purpose of this task is to examine trigonometric functions for obtuse angles. The values sinx and cosx are defined for acute angles by referring to a right triangle one of whose acute angles measures x. For an obtuse angle, no such triangle exists and so an alternate definition is required. Prior to working on this task, students should have experience working with trigonometric functions and how they relate to the unit circle. - What exactly is a radian?

Radians are often mysterious to students, yet they are a very straight forward way to measure an angle by relating the measure of the angle to the length of the arc on the unit circle it subtends. This task is not designed to discover the definition of radian, rather it allows students to make meaning out of the definition.

- As the Wheel Turns

In this task, students use trigonometric functions to model the movement of a point around a wheel and, in the case of part (c), through space (F-TF.5). Students also interpret features of graphs in terms of the given real-world context (F-IF.4). - Foxes and Rabbits 2

The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation. - Foxes and Rabbits 3

The example of rabbits and foxes was introduced to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.

- Global Positioning System II

Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. - Tennis Balls in a Can

This task is inspired by the derivation of the volume formula for the sphere.

- A Ton of Snow

The goal of this task is to examine a mathematical statement about the mass of snow, hopefully providing some stimulating thought to go along with the very arduous and demanding physical exercise. - Archimedes and the King's crown

This problem combines the ideas of ratio and proportion within the context of density of matter. - Eratosthenes and the circumference of the earth

This task is designed for the student to apply geometric concepts in modeling situations. - Framing a House - student task

This task has students recreate house plans on graph paper and then determine how many linear feet of wall plate material will be needed. - Global Positioning System II

Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. - Hexagonal Pattern of Beehives

The goal of this task is to use geometry study the structure of beehives. - How far is the horizon?

The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make reasonable assumptions and find information that is not provided in the task statement. - How many cells are in the human body?

The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context. - How many leaves on a tree?

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree. - How many leaves on a tree? (Version 2)

Teachers who use this version of the task will need to bring tree leaves (or prepare a good sketch of a tree leaf) to class so that they can work on and discuss how to approximate the area of an irregular shape like a leaf. - How thick is a soda can? Variation I

This task's main goal is to provide a familiar context and a straightforward question which require a variety of tools to solve: modeling a situation with geometry, paying close attention to units, and converting units. - How thick is a soda can? Variation II

This is a variation of ''How thick is a soda can? Variation I'' which allows students to work independently and think about how they can determine how thick a soda can is. - Ice Cream Cone

This rich task is an excellent example of geometric concepts in a modeling situation and is accessible to all students. In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper. - Indiana Jones and the Golden Statue

The goal of this task is to provide a introduction to the sometimes subtle use of density and units related to density, in a simple and fun context with minimal geometric complexity. - Paper Clip

This high level task is an example of applying geometric methods to solve design problems and satisfy physical constraints. - Regular Tessellations of the plane

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible. - Running around a track I

This task uses geometry to find the perimeter of the track. - Running around a track II

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. - Satellite

This task is an example of applying geometric methods to solve design problems and satisfy physical constraints. - Solar Eclipse

Total solar eclipses are quite rare, so much so that they make the news when they do occur. This task explores some of the reasons why. - Solving Quadratic Equations: Cutting Corners

This lesson unit is intended to help educators assess how well students are able to solve quadratics in one variable. - Tennis Balls in a Can

This task is inspired by the derivation of the volume formula for the sphere. - The Lighthouse Problem

In addition to the purely geometric and trigonometric aspects of the task, this problem asks students to model phenomena on the surface of the earth. - Tilt of earth's axis and the four seasons

This task gives students a chance to relate their weather experiences with a simple geometric model which explains why the seasons occur. - Toilet Roll

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints. - Use Cavalieris Principle to Compare Aquarium Volumes

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

- Accuracy of Carbon 14 Dating I

This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the statistical nature of such dating. - Do You Fit In This Car?

This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages. - Musical Preferences

The basic idea is for students to demonstrate that they know what it means for two variables to be associated. - SAT Scores

This task is designed to help students use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. They should also recognize that there are data sets for which such a procedure is not appropriate, and use calculators, spreadsheets, and tables to estimate areas under the normal curve. - Should We Send Out a Certificate?

The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions. The task is designed to encourage students to communicate their findings in a narrative/report form in context not just simply as a computed number.

- Fred's Flare Formula

This task is intended to engage students into considering how margin of error can be estimated from examining the results of repeated simple random sampling. - High blood pressure

The purpose of this task is to assess understanding of how study design dictates whether a conclusion of causation is warranted. - Musical Preferences

The basic idea is for students to demonstrate that they know what it means for two variables to be associated. - School Advisory Panel

This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - Strict Parents

This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - The Marble Jar

This task is designed as an instructional task to develop students understanding of how data from a random sample can be used to estimate a population proportion or percentage. - Types of Statistical Studies

The purpose of this task is to provide students with experience distinguishing between the various types of statistical studies and to understand the purpose of random selection in surveys and observational studies vs. random assignment to treatments in experiments. - Why Randomize?

This task is designed to help students understand statistics as a process for making inferences about population parameters based on a random sample from that population. - Words and Music II

The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of random assignment to experimental groups in an experiment.

- Fred's Flare Formula

This task is intended to engage students into considering how margin of error can be estimated from examining the results of repeated simple random sampling. - High blood pressure

The purpose of this task is to assess understanding of how study design dictates whether a conclusion of causation is warranted. - Margin of Error for Estimating a Population Mean

The purpose of this task is to illustrate the development of margin of error when estimating a population mean. - Scratch 'n Win Blues

This task is intended to engage students in considering how margin of error can be estimated from examining the results of repeated simple random sampling. - The Marble Jar

This task is designed as an instructional task to develop students understanding of how data from a random sample can be used to estimate a population proportion or percentage. - Types of Statistical Studies

The purpose of this task is to provide students with experience distinguishing between the various types of statistical studies and to understand the purpose of random selection in surveys and observational studies vs. random assignment to treatments in experiments. - Words and Music II

The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of random assignment to experimental groups in an experiment.

- Random Walk IV

This task completes the line of reasoning of Random Walk III in a situation where the numbers become too large to calculate and so abstract reasoning is required in order to compare the different probabilities. It is intended for instructional purposes only with a goal of understanding how to calculate and compare the combinatorial symbols.

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