Question 1

Maria, Daniel, Stephanie, Michael, Elena, Tyler, Sue, and Dimitri have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if Maria
And Daniel are to sit next to each other?

Accepted Answer

Maria and Daniel can be treated as a single unit, so there are 7 units to arrange. The number of ways to arrange 7 units is 7! (5040). Within their unit, Maria and Daniel can switch places, so there are 2! (2) ways to arrange them. Therefore, the total number of arrangements is 7! * 2! = 5040 * 2 = 10,080.

Question 2

Provide an appropriate response.
-Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books?
To solve this problem which of the following rules would you use?

Accepted Answer

The combinations rule is used because the order in which Allison picks the books does not matter; only the selection of three books from eight is important, making it a combination problem.

Question 3

Provide an appropriate response.
-Consider the following counting problem. Eight women and seven men are waiting in line at a movie theater. How many ways are there to arrange these 15 people amongst themselves such that
The eight women occupy the first eight places and the seven men the last seven places?
To solve this problem, which of the following rules would you use?

Accepted Answer

The permutations rule is used to arrange the eight women in the first eight places and the seven men in the last seven places. The fundamental counting principle is then applied to combine these two arrangements into one overall arrangement.

Question 4

Provide an appropriate response.
-Consider the selection of a nominating committee for a club. Is this a combination, a permutation, or neither?

Accepted Answer

The selection of a nominating committee for a club is a combination because the order in which the committee members are selected does not matter.

Question 5

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit
Together?

Accepted Answer

First we consider the women. Since all women must sit together, we can consider them as one group. Then, we have five groups: the group of women and each of the four groups of men. We can arrange these groups in $5!$ ways. Within the group of women, the women can arrange themselves in $4!$ ways. Similarly, within each group of men, the men can arrange themselves in $4!$ ways. Therefore, the total number of ways is $5! \cdot 4! \cdot 4! = 1,152$.

Question 6

Provide an appropriate response.
-Consider the selection of officers for a club. Is this a combination, a permutation, or neither?

Accepted Answer

The selection of officers for a club is a permutation because the order in which the officers are selected matters (e.g., president, vice president, secretary).

Question 7

How many 10-digit telephone numbers (area code + number)are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Accepted Answer

The first digit can be any digit from 1 to 9 (9 options), the next two digits can be any digit from 0 to 9 (10 options each), but we must exclude the combinations that form 800 and 900, which are 2 out of 1000 possible combinations for the first three digits. So, for the first three digits, we have 9 * 10 * 10 - 2 = 898 options. The next six digits are fixed as the number must end in 0000, leaving us with 898 * 1 = 898,000 possible telephone numbers.

Question 8

Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain all black cards.

Accepted Answer

The number of ways to choose 5 black cards out of 26 black cards (spades and clubs) in a deck is calculated using the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose. Here, n=26 (total black cards) and k=5 (cards to choose), so C(26, 5) = 26! / [5!(26-5)!] = 65,780.

Question 9

Provide an appropriate response.
-Consider the selection of seven apples from a barrel of 75 apples. Is this a combination, a permutation, or neither?

Accepted Answer

Selecting seven apples from a barrel of 75 without regard to the order in which they are selected is a combination, as the order of selection does not matter.

Question 10

Provide an appropriate response.
-Consider determining how many possible phone numbers are within a particular area code (repeated numbers allowed). Is this a combination, a permutation, or neither?

Accepted Answer

This is neither a combination nor a permutation because it involves creating sequences where repetition is allowed and order matters, which is different from the criteria defining combinations and permutations.

Question 11

A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four hearts and one spade?

Accepted Answer

To form a hand with four hearts and one spade, choose 4 hearts from the 13 available hearts and 1 spade from the 13 available spades. The number of ways to choose 4 hearts is C(13,4) = 715, and the number of ways to choose 1 spade is C(13,1) = 13. Multiply these together: 715 * 13 = 9295.

Question 12

A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four cards of one suit and one card of another suit?

Accepted Answer

The answer of A poker hand consists of 5 cards...

Question 13

A committee is to be chosen from a group of fourteen women and twelve men. Determine the number of ways of choosing a female president, a male treasurer, and three other members of
Either gender. Assume that no one can hold more than one office.

Accepted Answer

The answer of A committee is to be chosen from...

Question 14

Provide an appropriate response.
-Consider the following counting problem. How many different sequences of 3 letters can be formed using the letters a, b, c, d, e if repetition is allowed?
To solve this problem which of the following rules would you use?

Accepted Answer

The fundamental counting principle is used here because for each position in the sequence (3 positions in total), any of the 5 letters can be chosen, allowing for repetition.

Question 15

How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 8 members? Assume that no member can hold more than one office.

Accepted Answer

This is a permutation problem because the order in which the officers are chosen matters. There are 8 options for president, 7 remaining options for vice-president, 6 for secretary, and 5 for treasurer. Thus, the total number of ways is 8 * 7 * 6 * 5 = 1680.

Question 16

In how many ways could a group of fourteen people be divided into four groups containing respectively two, three, four, and five people?

Accepted Answer

The number of ways to divide 14 people into groups of 2, 3, 4, and 5 is calculated using the multinomial coefficient: 14! / (2! * 3! * 4! * 5!). This evaluates to 2,522,520.

Question 17

License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

Since repetition is allowed, each of the 3 letters can be any of the 26 letters in the alphabet, and each of the 3 digits can be any of the 10 digits (0-9). Therefore, the total number of plates is $26^3 \times 10^3 = 17,576,000$.

Question 18

Provide an appropriate response.
-Consider the arrangements of the letters in the word ALGEBRA. Is this a combination, a permutation, or neither?

Accepted Answer

Arrangements of letters consider the order, making it a permutation.

Question 19

Provide an appropriate response.
-Consider the following counting problem. A pool of possible jurors consists of 11 men and 13 women. How many different juries consisting of 5 women and 7 men are possible?
To solve this problem, which of the following rules would you use?

Accepted Answer

To form a jury of 5 women and 7 men, you use the combinations rule to choose 5 women from 13 and 7 men from 11, then apply the fundamental counting principle to combine these choices.

Question 20

Provide an appropriate response.
-Consider the selection of a thirteen card bridge hand. Is this a combination, a permutation, or neither?

Accepted Answer

In a bridge hand, the order in which the cards are selected does not matter, only the specific cards that are selected. This is the definition of a combination.

Quiz 10: Counting Methods

Maria, Daniel, Stephanie, Michael, Elena, Tyler, Sue, and Dimitri have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if Maria And Daniel are to sit next to each other?

Provide an appropriate response. -Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books? To solve this problem which of the following rules would you use?

Provide an appropriate response. -Consider the selection of a nominating committee for a club. Is this a combination, a permutation, or neither?

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit Together?

Provide an appropriate response. -Consider the selection of officers for a club. Is this a combination, a permutation, or neither?

How many 10-digit telephone numbers (area code + number) are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain all black cards.

Provide an appropriate response. -Consider the selection of seven apples from a barrel of 75 apples. Is this a combination, a permutation, or neither?

Provide an appropriate response. -Consider determining how many possible phone numbers are within a particular area code (repeated numbers allowed) . Is this a combination, a permutation, or neither?

A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four hearts and one spade?

A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four cards of one suit and one card of another suit?

A committee is to be chosen from a group of fourteen women and twelve men. Determine the number of ways of choosing a female president, a male treasurer, and three other members of Either gender. Assume that no one can hold more than one office.

Provide an appropriate response. -Consider the following counting problem. How many different sequences of 3 letters can be formed using the letters a, b, c, d, e if repetition is allowed? To solve this problem which of the following rules would you use?

How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 8 members? Assume that no member can hold more than one office.

In how many ways could a group of fourteen people be divided into four groups containing respectively two, three, four, and five people?

License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Provide an appropriate response. -Consider the arrangements of the letters in the word ALGEBRA. Is this a combination, a permutation, or neither?

Provide an appropriate response. -Consider the following counting problem. A pool of possible jurors consists of 11 men and 13 women. How many different juries consisting of 5 women and 7 men are possible? To solve this problem, which of the following rules would you use?

Provide an appropriate response. -Consider the selection of a thirteen card bridge hand. Is this a combination, a permutation, or neither?