Section 5.4.H: Homework
- Page ID
- 215611
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Section 5.4: Combinations - Homework Exercises
A club has 8 members. They need to select 3 members to form a planning committee. How many different committees are possible?
- Answer
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\(_8C_3=\) 56 different committees
A pizza shop offers 10 different toppings. You want to order a pizza with exactly 4 toppings. How many different 4-topping pizzas can you create?
- Answer
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\(_{10}C_4=\) 210 different pizzas
A basketball team has 12 players. The coach needs to choose 2 players to be team captains. How many different pairs of captains are possible?
- Answer
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\(_{12}C_2=\) 68 different pairs
You have 15 books on your shelf and are going on vacation. You can only pack 5 books. How many different sets of 5 books could you choose?
- Answer
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\(_{15}C_5=\) 3,003 different sets of books
A company has 8 men and 7 women. They need to form a 5-person committee with exactly 3 men and 2 women. How many different committees are possible?
- Answer
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\((_8C_3)\cdot\left(_7C_2\right)=56 \cdot 21=\text{1,176}\) different committees
An ice cream shop has 8 fruit flavors and 6 chocolate-based flavors. You want to choose 3 scoops with 2 fruit flavors. How many combinations are possible?
- Answer
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\((_8C_4)\cdot\left(_6C_3\right)=70\cdot20=\text{1,400}\) different combinations
A debate team has 10 members, including their captain Sarah. They need to choose 4 people to compete in a tournament, and Sarah must be one of them. How many different 4-person teams are possible?
- Answer
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\(_{10}C_4=\) 210 different teams
A jury pool consists of 18 people: 10 women and 8 men. How many juries can be formed with 7 women and 5 men?
- Answer
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\((_{10}C_7)\cdot\left(_8C_5\right)=20\cdot56=\text{1,120}\) different juries
A school has 25 freshmen, 30 sophomores, 28 juniors, and 22 seniors. They need to select a 6-person student council with at least one student from each class. How many different councils are possible if they select exactly 2 freshmen, 1 sophomore, 2 juniors, and 1 senior?
- Answer
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\((_{25}C_2)\cdot\left(_{30}C_1\right)\cdot\left(_{28}C_2\right)\cdot\left(_{22}C_1\right)=300\cdot30\cdot378\cdot22=\text{74,844,000}\) different councils
Powerball is a two-drum lottery game, meaning winning numbers are drawn from two separate machines. Powerball is an American lottery game that is played in 45 States, Washington D.C., Puerto Rico, and the US Virgin Islands. (As of 2025, not played in Nevada, Utah, Alabama, Alaska, and Hawaii). The current version of Powerball is as follows: It costs $2.00 to play. You are to pick five numbers numbered from 1–69 and one number (“The Powerball”) numbered 1–26. Every Wednesday and Saturday night at 10:59 p.m. Eastern Time, the five white balls out of a drum with 69 balls and the one red “Powerball” out of a drum with 26 red balls will be selected and revealed. The selection process is random and each of the 6-balls chosen has the same likeliness of being drawn. (On November 7, 2022, Powerball produced the largest lottery jackpot in history; the $2.04 billion jackpot was won by someone in Altadena, California.)
Here are the payouts along with the odds of winning based on the winning combinations:
- In how many ways can someone choose 5 white balls and 1 red powerball?
- In how many ways can someone win by matching the 5 white balls and 1 red powerball?
- Answer
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- \((_{69}C_5)\cdot\left(_{26}C_1\right)=11,238,513\cdot26=\) 298,201,338 different choices
- \((_5C_5)\cdot\left(_1C_1\right)=1\cdot1=1\) way to win