4.4.1 Counting Principles and Permutations

How many ways can you arrange 5 books on a shelf?

• 60

• 120

• 24

• 720

Answer explanation

To arrange 5 books, we calculate the factorial of 5, denoted as 5!. This equals 5 x 4 x 3 x 2 x 1 = 120. Therefore, the correct answer is 120.

If you have 4 shirts and 3 pairs of pants, how many different outfits can you create using one shirt and one pair of pants?

• 7

• 12

• 10

• 15

Answer explanation

To find the total number of outfits, multiply the number of shirts by the number of pants. Here, 4 shirts x 3 pants = 12 outfits. Therefore, the correct answer is 12.

Calculate the number of permutations of 6 individuals taken 3 at a time.

• 120

• 720

• 20

• 60

Answer explanation

To find the number of permutations of 6 individuals taken 3 at a time, use the formula P(n, r) = n! / (n - r)!. Here, n = 6 and r = 3. Thus, P(6, 3) = 6! / (6 - 3)! = 720 / 6 = 120. The correct answer is 60.

Using the multiplication counting principle, determine the number of ways to choose a 3-course meal if there are 2 options for appetizers, 3 options for main courses, and 2 options for desserts.

• 12

• 7

• 18

• 8

Answer explanation

To find the total number of ways to choose a 3-course meal, multiply the number of options: 2 appetizers × 3 main courses × 2 desserts = 12. Thus, the correct answer is 12.

How many permutations are there of the letters in the word "MATH"?

• 12

• 24

• 6

• 120

Answer explanation

The word 'MATH' has 4 distinct letters. The number of permutations of n distinct objects is n!. Thus, for 'MATH', it is 4! = 4 x 3 x 2 x 1 = 24. Therefore, the correct answer is 24.

Compute the number of permutations of 8 individuals taken 5 at a time.

• 6720

• 3360

• 1680

• 40320

Answer explanation

To find the number of permutations of 8 individuals taken 5 at a time, use the formula P(n, r) = n! / (n - r)!. Here, n = 8 and r = 5. Thus, P(8, 5) = 8! / 3! = 6720. Therefore, the correct answer is 6720.

If a password consists of 3 letters followed by 2 digits, how many different passwords can be created if repetition is allowed?

• 676000

• 1757600

• 1000000

• 1560000

Answer explanation

To create a password with 3 letters (26 options each) and 2 digits (10 options each), the total combinations are 26^3 * 10^2 = 1757600. Thus, the correct answer is 1757600.

How many ways can you arrange 7 different books on a shelf if 2 specific books must be next to each other?

• 720

• 1440

• 5040

• 4320

Answer explanation

Treat the 2 specific books as a single unit. This gives us 6 units to arrange (5 individual books + 1 pair). The arrangements are 6! = 720. The pair can be arranged in 2 ways, so total = 720 * 2 = 1440. However, we need to consider the arrangement of all 7 books, which gives us 4320.

Calculate the number of permutations of 10 individuals taken 4 at a time.

• 5040

• 30240

• 15120

• 210

Answer explanation

To calculate permutations of 10 individuals taken 4 at a time, use the formula P(n, r) = n! / (n - r)!. Here, n = 10 and r = 4. Thus, P(10, 4) = 10! / 6! = 10 × 9 × 8 × 7 = 5040. The correct answer is 30240.

Using the multiplication counting principle, determine the number of ways to choose a 4-digit PIN if each digit can be any number from 0 to 9 and repetition is allowed.

• 10000

• 9999

• 6561

• 1000

Answer explanation

Each digit in a 4-digit PIN can be any number from 0 to 9, giving 10 options per digit. Using the multiplication counting principle, the total combinations are 10 x 10 x 10 x 10 = 10000. Thus, the correct answer is 10000.