Question 1

Solve the problem.
-How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Accepted Answer

The first letter has 2 choices (K or W), the next two letters have 26 choices each (A-Z), and the last letter has 25 choices (A-Z except O). So, the total number of combinations is 2 * 26 * 26 * 25 = 33,800.

Question 2

Solve the problem.
-License plates are made using 2 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

There are 26 choices for each letter and 10 choices for each digit. Therefore, the total number of plates is 26 * 26 * 10 * 10 = 67,600.

Question 3

Solve the problem.
-A local department store sells carpets in 4 sizes. Each carpet comes in 3 different qualities. One of the sizes comes in 5 colors. The other sizes come in 3 colors. How many choices of carpet are there?

Accepted Answer

The answer of Solve the problem.
-A local department store sells...

Question 4

Solve the problem.
-How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

Accepted Answer

The first digit has 3 choices (3, 4, or 5). The next two digits each have 10 choices (0-9). The last digit has 999 choices (000 to 999, excluding 000). So, the total number of sequences is 3 * 10 * 10 * 9 = 2,997.

Question 5

Solve the problem.
-How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?

Accepted Answer

A 5-digit number can be formed by choosing each digit from 0 to 9. The first digit cannot be 0 (to ensure it's a 5-digit number), so there are 9 choices for the first digit and 10 choices for each of the remaining four digits. Thus, the total number of 5-digit numbers is 9 * 10^4 = 90,000.

Question 6

Solve the problem.
-A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. How many different
Combinations did he have?

Accepted Answer

There are 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. So, the total number of combinations is 6 + 5 + 3 = 14.

Question 7

Solve the problem.
-Given a committee of 8 women and 11 men, count the number of different ways of choosing a president, a secretary, and a treasurer, if the president must be a woman and the secretary and
Treasurer must be men. Assume no one can hold more than one office.

Accepted Answer

The answer of Solve the problem.
-Given a committee of 8...

Question 8

Solve the problem.
-Mark can remember only the first 4 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0. If Mark were to dial all of the possible numbers
And if it takes him 21 seconds to try each one, how long would it take to try every possibility?

Accepted Answer

Since Mark remembers the first 4 digits of a 7-digit number and knows the last digit is not 0, he needs to guess the remaining 2 digits and the last digit with the constraint. For the 2 unknown digits, each can be any number from 0-9, giving 10 possibilities each. For the last digit, it can be any number from 1-9 (10 possibilities minus the 0), giving 9 possibilities. Thus, the total number of combinations is 10 * 10 * 9 = 900. At 21 seconds per attempt, the total time is 900 * 21 = 18900 seconds. Converting seconds to minutes, 18900 seconds / 60 = 315 minutes.

Question 9

Solve the problem.
-A restaurant offers salads with 2 types of lettuce, 5 different toppings, and 5 different dressings. How many different salads could be ordered?

Accepted Answer

To find the total number of different salads, multiply the number of choices for each component: 2 types of lettuce * 5 toppings * 5 dressings = 50 different salads.

Question 10

Solve the problem.
-In how many ways can 6 people line up for play tickets?

Accepted Answer

The number of ways 6 people can line up for play tickets is calculated using the formula for permutations, which is n!, where n is the number of people. So, for 6 people, it's 6! = 6*5*4*3*2*1 = 720.

Question 11

Solve the problem.
-How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

Accepted Answer

The number of ways to choose a president, vice-president, and secretary from 12 members is a permutation of 12 items taken 3 at a time, which is calculated as 12P3 = 12! / (12-3)! = 12*11*10 = 1320.

Question 12

Solve the problem.
-At a lumber company, shelves are sold in 4 types of wood, 2 different widths and 5 different lengths. How many different types of shelves could be ordered?

Accepted Answer

The total number of different types of shelves is found by multiplying the number of options for each characteristic: 4 types of wood * 2 widths * 5 lengths = 40 different types of shelves.

Question 13

Solve the problem.
-A shoe store carries one brand of shoe in 4 different styles, 5 sizes, and 5 colors. How many different shoes are available of this one brand?

Accepted Answer

The total number of different shoes available is calculated by multiplying the number of styles, sizes, and colors together: 4 styles * 5 sizes * 5 colors = 100 different shoes.

Question 14

Solve the problem.
-A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How
Many different combinations did she have?

Accepted Answer

The saleswoman could create combinations as follows: with the first jacket, she can pair all 6 skirts, giving 6 combinations. With the second jacket, she can pair 5 skirts, adding 5 more combinations. With the third jacket, she can pair 2 skirts, adding 2 more combinations. In total, this gives 6 + 5 + 2 = 13 different combinations.

Question 15

Solve the problem.
-A sports shop sells tennis rackets in 4 different weights, 3 types of string, and 3 grip sizes. How many different rackets could they sell?

Accepted Answer

The total number of different rackets is calculated by multiplying the number of options for each feature: 4 weights * 3 types of string * 3 grip sizes = 36 different rackets.

Question 16

Solve the problem.
-If 5 newborn babies are randomly selected, how many different gender sequences are possible?

Accepted Answer

Each baby can be either male or female, so there are 2 possibilities for each of the 5 babies. Thus, the total number of different gender sequences is $2^5 = 32$.

Question 17

Solve the problem.
-Given a group of students: $\mathrm { G } = \{$ Allen, Brenda, Chad, Dorothy, Eric, Frances, Gale $\}$, count the number of different ways of choosing 4 people for a committee. Assume no one can hold more than one office and that each person is to hold a different position on the committee.

Accepted Answer

The number of ways to choose 4 people out of 7 for different positions is calculated using permutations since the order matters. The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$. Here, $n = 7$ and $k = 4$, so $P(7, 4) = \frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4}{1} = 840$.

Question 18

Solve the problem.
-Given a group of 8 women and 11 men, how many different ways are there of choosing one man and one woman for a committee?

Accepted Answer

There are 8 ways to choose a woman and 11 ways to choose a man. Therefore, the total number of ways to choose one man and one woman is 8 * 11 = 88.

Question 19

Solve the problem.
-A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

Accepted Answer

The answer of Solve the problem.
-A baseball manager has 10...

Question 20

Solve the problem.
-A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

Accepted Answer

The number of ways to arrange 5 selections out of 7 is calculated using permutations since the order of the selections matters. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to choose, and "!" denotes factorial. Here, n=7 and r=5, so P(7, 5) = 7! / (7-5)! = 7! / 2! = (7×6×5×4×3) / (2×1) = 2520.

Quiz 10: Counting Methods

Solve the problem. -How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Solve the problem. -License plates are made using 2 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Solve the problem. -A local department store sells carpets in 4 sizes. Each carpet comes in 3 different qualities. One of the sizes comes in 5 colors. The other sizes come in 3 colors. How many choices of carpet are there?

Solve the problem. -How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

Solve the problem. -How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?

Solve the problem. -A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. How many different Combinations did he have?

Solve the problem. -Given a committee of 8 women and 11 men, count the number of different ways of choosing a president, a secretary, and a treasurer, if the president must be a woman and the secretary and Treasurer must be men. Assume no one can hold more than one office.

Solve the problem. -A restaurant offers salads with 2 types of lettuce, 5 different toppings, and 5 different dressings. How many different salads could be ordered?

Solve the problem. -In how many ways can 6 people line up for play tickets?

Solve the problem. -How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

Solve the problem. -At a lumber company, shelves are sold in 4 types of wood, 2 different widths and 5 different lengths. How many different types of shelves could be ordered?

Solve the problem. -A shoe store carries one brand of shoe in 4 different styles, 5 sizes, and 5 colors. How many different shoes are available of this one brand?

Solve the problem. -A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How Many different combinations did she have?

Solve the problem. -A sports shop sells tennis rackets in 4 different weights, 3 types of string, and 3 grip sizes. How many different rackets could they sell?

Solve the problem. -If 5 newborn babies are randomly selected, how many different gender sequences are possible?

Solve the problem. -Given a group of 8 women and 11 men, how many different ways are there of choosing one man and one woman for a committee?

Solve the problem. -A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

Solve the problem. -A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?