Question 1

Determine whether the statement is true or false.
​
If there are eight toppings available, then the number of different pizzas that can be made is  , or 4, different pizzas.

Accepted Answer

The number of different pizzas that can be made with eight toppings is 2^8, which equals 256.

Question 2

How many different batting orders can be formed for a six-member baseball team? ​

Accepted Answer

The answer of How many different batting orders can be...

Question 3

How many eight-letter permutations can be formed from the first letters of the alphabet? ​

Accepted Answer

The answer of How many eight-letter permutations can be formed...

Question 4

In how many ways can an investor select two mutual funds for his investment portfolio from a recommended list of nine mutual funds? ​

Accepted Answer

The answer of In how many ways can an investor...

Question 5

In how many ways can six students be seated in a row of six seats? ​

Accepted Answer

The answer of In how many ways can six students...

Question 6

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Question 7

How many six-letter permutations can be formed from the first six letters of the alphabet? ​

Accepted Answer

There are 6 options for the first letter, 5 for the second, 4 for the third, and so on until only 1 option remains for the last letter. The total number of permutations is therefore 6 × 5 × 4 × 3 × 2 × 1 = 720.

Question 8

An exam consists of five true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam?
​
__________
​
In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered?
​
__________

Accepted Answer

The answer of An exam consists of five true-or-false questions....

Question 9

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Question 10

In how many ways can a supermarket chain select 7 out of 17 possible sites for the construction of new supermarkets? ​

Accepted Answer

The answer of In how many ways can a supermarket...

Question 11

In how many ways can a quality-control engineer select a sample of 5 transistors for testing from a batch of 90 transistors? ​

Accepted Answer

The answer of In how many ways can a quality-control...

Question 12

In how many ways can a member of a hiring committee select 6 of 17 job applicants for further consideration? ​

Accepted Answer

This is a combination problem, where the order of the selected applicants doesn't matter. Therefore, we can use the combination formula: 
nCk = n! / (k!(n-k)!)
Plugging in the values, we get: 
17C6 = 17! / (6!(17-6)!) = 17! / (6!11!) = 24,887,149 
Therefore, there are 24,887,149 ways to select 6 job applicants from a pool of 17 for further consideration.

Question 13

UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts? ​

Accepted Answer

Since no firm can receive more than two contracts, the three contracts can be divided in four ways: 
1. One firm receives two contracts, two firms receive one contract each. There are 12 options for selecting the firm receiving two contracts, and 11 options for selecting the firms receiving one contract each (after the first firm has been selected). The total number of possibilities is 12*11*10. 
2. Two firms receive two contracts each, one firm receives one contract. There are 12 options for selecting the firm receiving one contract, and 11 options for selecting the first firm receiving two contracts (after the first firm has been selected). For each of these options, there are 10 options for selecting the second firm receiving two contracts (after the first firm has been selected). The total number of possibilities is 12*11*10. 
3. Three firms receive one contract each. There are 12 options for selecting the firm receiving the first contract, 11 options for selecting the firm receiving the second contract (after the first firm has been selected), and 10 options for selecting the firm receiving the third contract. The total number of possibilities is 12*11*10. 
4. One firm receives one contract, two firms receive two contracts each. There are 12 options for selecting the firm receiving one contract, and 11 options for selecting the first firm receiving two contracts (after the first firm has been selected). For each of these options, there are 10 options for selecting the second firm receiving two contracts (after the first firm has been selected). The total number of possibilities is 12*11*10.

Therefore, the total number of ways the contracts can be awarded among these firms is: 
12*11*10 + 12*11*10 + 12*11*10 + 12*11*10 = 5280

Therefore, C is the correct answer.

Question 14

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Question 15

A student planning her curriculum for the upcoming year must select one of six Business courses, one of seven Mathematics courses, two of three elective courses, and either one of four History courses or one of seven Social Science courses. How many different curricula are available for her consideration? ​

Accepted Answer

To calculate the total number of possible curricula, we need to multiply the number of choices for each category. Therefore, the total number of curricula is:
6 x 7 x (3 choose 2) x (4 + 7) = 42 x 3 x 11 = 1386
So, the correct answer is B, which matches this calculation.

Question 16

Determine whether the statement is true or false.
​
The number of four-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 179.

Accepted Answer

To form an even number, the units digit must be 2 or 4. 
Case 1: Units digit is 2 
Number of choices for units digit = 1 
Number of choices for thousands digit = 3 (cannot be 1) 
Number of choices for hundreds digit = 2 (cannot be 2 or 4) 
Number of choices for tens digit = 1 (must be 4) 
Total number of choices = 1 × 3 × 2 × 1 = 6 
Case 2: Units digit is 4 
Number of choices for units digit = 1 
Number of choices for thousands digit = 3 (cannot be 1) 
Number of choices for hundreds digit = 2 (cannot be 2 or 4) 
Number of choices for tens digit = 1 (must be 2) 
Total number of choices = 1 × 3 × 2 × 1 = 6 
Total number of four-digit even numbers = 6 + 6 = 12. Therefore, the statement is false.

Question 17

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Question 18

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Question 19

How many three-letter permutations can be formed from the first five letters of the alphabet? ​

Accepted Answer

There are 5 choices for the first letter, 4 choices for the second letter (since one letter has already been picked), and 3 choices for the third letter. This gives us a total of 5 x 4 x 3 = 60 permutations.

Question 20

Evaluate the expression. ​  ​

Accepted Answer

The answer of Evaluate the expression. ​  ​
...

Quiz 6: Sets and Counting

Determine whether the statement is true or false. If there are eight toppings available, then the number of different pizzas that can be made is , or 4, different pizzas.

How many different batting orders can be formed for a six-member baseball team?

How many eight-letter permutations can be formed from the first letters of the alphabet?

In how many ways can an investor select two mutual funds for his investment portfolio from a recommended list of nine mutual funds?

In how many ways can six students be seated in a row of six seats?

Evaluate the expression.

How many six-letter permutations can be formed from the first six letters of the alphabet?

An exam consists of five true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered?

Evaluate the expression.

In how many ways can a supermarket chain select 7 out of 17 possible sites for the construction of new supermarkets?

In how many ways can a quality-control engineer select a sample of 5 transistors for testing from a batch of 90 transistors?

In how many ways can a member of a hiring committee select 6 of 17 job applicants for further consideration?

UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts?

Evaluate the expression.

Determine whether the statement is true or false. The number of four-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 179.

Evaluate the expression.

Evaluate the expression.

How many three-letter permutations can be formed from the first five letters of the alphabet?

Evaluate the expression.

Quiz 6: Sets and Counting

Determine whether the statement is true or false. ​ If there are eight toppings available, then the number of different pizzas that can be made is , or 4, different pizzas.

How many different batting orders can be formed for a six-member baseball team? ​

How many eight-letter permutations can be formed from the first letters of the alphabet? ​

In how many ways can an investor select two mutual funds for his investment portfolio from a recommended list of nine mutual funds? ​

In how many ways can six students be seated in a row of six seats? ​

Evaluate the expression. ​ ​

How many six-letter permutations can be formed from the first six letters of the alphabet? ​

Evaluate the expression. ​ ​

In how many ways can a supermarket chain select 7 out of 17 possible sites for the construction of new supermarkets? ​

In how many ways can a quality-control engineer select a sample of 5 transistors for testing from a batch of 90 transistors? ​

In how many ways can a member of a hiring committee select 6 of 17 job applicants for further consideration? ​

UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts? ​

Evaluate the expression. ​ ​

Determine whether the statement is true or false. ​ The number of four-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 179.

Evaluate the expression. ​ ​

Evaluate the expression. ​ ​

How many three-letter permutations can be formed from the first five letters of the alphabet? ​

Evaluate the expression. ​ ​

Determine whether the statement is true or false. If there are eight toppings available, then the number of different pizzas that can be made is , or 4, different pizzas.

How many different batting orders can be formed for a six-member baseball team?

How many eight-letter permutations can be formed from the first letters of the alphabet?

In how many ways can an investor select two mutual funds for his investment portfolio from a recommended list of nine mutual funds?

In how many ways can six students be seated in a row of six seats?

Evaluate the expression.

How many six-letter permutations can be formed from the first six letters of the alphabet?

Evaluate the expression.

In how many ways can a supermarket chain select 7 out of 17 possible sites for the construction of new supermarkets?

In how many ways can a quality-control engineer select a sample of 5 transistors for testing from a batch of 90 transistors?

In how many ways can a member of a hiring committee select 6 of 17 job applicants for further consideration?

UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts?

Evaluate the expression.

Determine whether the statement is true or false. The number of four-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 179.

Evaluate the expression.

Evaluate the expression.

How many three-letter permutations can be formed from the first five letters of the alphabet?

Evaluate the expression.