7.4.1: Exercises

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Section 7.3 Exercises

For the following exercises, decide whether the situation describes a permutation or a combination.

You’re packing for vacation, and you need to pick 5 shirts.

You and your friends are about to play a game, and you need to decide who will have the first turn, second turn, and so on.

You are watching your favorite reality show, and you want to know how many possibilities there are for the order of finish for the top three.

You are going to be working in groups of 4 with your classmates, and you want to know how many possibilities there are for the composition of your group.

For the following exercises, express your answers as whole numbers.

$_5{C_3}$

$_8{C_2}$

$_8{C_6}$

$_{12}{C_3}$

$_{12}{C_5}$

10.

$_{14}{C_3}$

11.

$_{14}{C_{10}}$

12.

$_{15}{C_5}$

13.

$_{15}{C_{13}}$

14.

$_{18}{C_3}$

15.

$_{18}{C_6}$

16.

$_{20}{C_4}$

17.

In most variations of the card game poker, a hand consists of 5 cards, where the order doesn’t matter. How many different poker hands are there?

18.

A professor starts each class by choosing 3 students to present solutions to homework problems to the class. If there are 41 students in the class, in how many different ways can the professor make those selections?

19.

An election for at-large members of a school board has 7 candidates; 3 will be elected. How many different ways can those 3 seats be filled?

20.

There are 20 contestants on a reality TV show; at the end of the first episode, 10 are eliminated. How many different groups of eliminated contestants are possible?

21.

At a horse race, bettors can place a bet called an exacta box. For this bet, the player chooses 2 horses; if those horses finish first and second (in either order), the player wins. In a race with 12 horses in the field, how many possible exacta box bets are there?

The following exercises are about the card game euchre, which uses a partial standard deck of cards: it only has the cards with ranks 9, 10, J, Q, K, and A (for a total of 24 cards). Some variations of the game use the 8s or the 7s and 8s, but we’ll stick with the 24-card version.

22.

A euchre hand contains 5 cards. How many ways are there to receive a 5-card hand (where the order in which the cards are received doesn’t matter, i.e., 9$\heartsuit$, J$\heartsuit$, ${\text{K}}\clubsuit $, $9\spadesuit $, $10\spadesuit $ is the same as $9\spadesuit $ J$\heartsuit$, 9$\heartsuit$, ${\text{K}}\clubsuit $, $10\spadesuit $)?

23.

After all 4 players get their hands, the remaining 4 cards are placed face down in the center of the table. How many different groups of 4 cards are there from this deck?

24.

Euchre is played with partners. How many ways are there for 2 partners to receive 5-card hands (where, as above, the order doesn’t matter)? Hint: After the first person gets their cards, there are $52 - 5 = 47$ cards left for the second person.

You and 5 of your friends are at an amusement park, and are about to ride a roller coaster. The cars have room for 6 people arranged in 3 rows of 2, so you and your friends will perfectly fill one car.

25.

How many ways are there to choose the 2 people in the front row?

26.

Assuming the front row has been selected, how many ways are there to choose the 2 people in the middle row?

27.

Assuming the first 2 rows have been selected, how many ways are there to choose the 2 people in the back row?

28.

Using the Multiplication Rule for Counting and your answers to the earlier parts of this exercise, how many ways are there for your friends to sort yourselves into rows to board the roller coaster?

The University Combinatorics Club has 18 members. Four of them will be selected to form a committee.

29.

How many different committees of 4 are possible, assuming all of the duties are shared equally?

30.

Instead of sharing responsibility equally, one person will be chosen to be the committee chair. How many different committees are possible? Count these by selecting a chair first, then selecting the remaining 3 members of the committee from the remaining club members and use the Multiplication Rule for Counting. Show your work.

31.

Let’s count the number of committees with chairs a different way: First, choose 4 people for the committee (as in the first question), then choose 1 of the 4 to be chair. Show your work. Do you get the same number?

Powerball^® is a multistate lottery game, which costs $2 to play. Players fill out a ticket by choosing 5 numbers between 1 and 69 (these are the white balls) and then a single number between 1 and 26 (this is the Powerball).

32.

How many different ways are there to choose the white balls? Players who match these 5 numbers exactly (but not the Powerball) win $1 million.

33.

How many ways are there to choose the Powerball? Players who correctly pick the Powerball win $4.

34.

How many ways are there to play the game altogether? Players who match all 5 white balls and the Powerball win (or share) the grand prize. (The grand prize starts at $40 million; if no players win the grand prize, the value goes up for the next drawing. The highest value it has ever reached is $1.586 billion!) How many ways are there to fill out a single Powerball ticket?

35.

You are in charge of programming for a music festival. The festival has a main stage, a secondary stage, and several smaller stages. There are 40 bands confirmed for the festival. Five of those will play the main stage, and 8 will play the secondary stage. How many ways are there for you to allocate bands to these 2 stages?

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