# Permutations and Combinations Worksheet - Page 4

Permutations and Combinations Worksheet
• Page 4
31.
What is the value of the expression $\frac{180}{{}_{5}{P}_{3}}$?
 a. 4 b. 3 c. 5 d. None of the above

#### Solution:

5P3 = 5×4×3 = 60
[Expand 5P3 .]

1805P3 = 180 / 60 = 3
[Substitute the value of 5P3 in the expression and simplify.]

The value of 1805P3 is 3.

32.
What is the value of 5!?
 a. 5 b. 124 c. 120 d. 11 e. 13

#### Solution:

The factorial of a number is the product of all the whole numbers from 1 to that number.

5! = 5 × 4 × 3 × 2 × 1
[Expand 5!.]

5! = 120
[Multiply.]

The value of 5! is 120.

33.
Find the number of ways of choosing 4 members from a team of 15 members.
 a. 179 b. 1365 c. 1265 d. 169

#### Solution:

Number of ways of choosing 4 members from 15 members = 15C4

15C4 = 1 / 4! × 15P4
[Expand 15C4.]

15P4 = 15 × 14 × 13 × 12
[Expand 15P4 .]

4! = 4 × 3 × 2 × 1 = 24
[Expand 4!.]

15C4 = 1 / 24 × (15 × 14 × 13 × 12)
[Substitute the values of 4! and 15P4 .]

= 15 × 7 × 13 = 1365
[Simplify.]

Number of ways of choosing 4 members from a team of 15 members = 1365

34.
What is the value of 2! + 5!?
 a. 127 b. 124 c. 122 d. None of the above

#### Solution:

2! = 2 ×1 = 2
[Expand 2! and multiply.]

5! = 5 ×4 ×3 ×2 ×1 = 120
[Expand 5! and multiply.]

2! + 5! = 2 + 120 = 122

The value of (2! + 5!) is 122.

35.
In how many ways can 7 books be arranged in a shelf?
 a. 8! b. 7C7 c. 77 d. 7!

#### Solution:

Number of books = 7

Number of ways in which 7 books can be arranged = 7P7

7P7 = 7!

There are 7! ways to arrange 7 books in a shelf.

36.
Evaluate: 4! × (1! + 0!)
 a. 53 b. 50 c. 48 d. 49

#### Solution:

4! = 4 ×3 ×2 ×1 = 24
[Expand 4! and multiply.]

1! + 0! = 1 + 1 = 2
[Substitute 1! = 1 and 0! = 1.]

4! x (1! + 0!) = 24 x 2 = 48
[Substitute the values of 4! and (1! + 0!).]

37.
What is the value of $\frac{9!}{\left(9-2\right)!}$?
 a. 75 b. 72 c. 66 d. 77 e. 144

#### Solution:

9! = 9 × 8 × 7!
[Expand 9!.]

(9 - 2)! = 7!
[Simplify.]

9!(9-2)! = 9 × 8 × 7!7!
[Substitute the values of 9! and (9 - 2)!.]

= 9 × 8 = 72
[Simplify.]

The value of 9! / (9-2)! is 72.

38.
Marissa has 5 different toys and wants to display 3 of them. In how many ways can they be selected and arranged?
 a. 30 b. 20 c. 55 d. 60

#### Solution:

Number of ways of selecting 3 out of 5 toys = 5C3

Number of ways in which 3 toys can be arranged = 3!

Number of ways in which 3 out of 5 toys can be selected and arranged = 5C3 × 3! = 5P3

= 5! / (5-3)! = 5! / 2! = 60
[Simplify.]

There are 60 different ways to select 3 of the 5 toys and arrange them for display.

39.
Find the number of arrangements of the letters of the word MATH.
 a. 12 b. 24 c. 9 d. 10

#### Solution:

The word 'MATH' has 4 letters.

There are 4 letters to put in the first place, 3 letters to put in the second place, 2 letters to put in the third place and 1 letter to put in the fourth place.

By counting principle, number of arrangements = 4 × 3 × 2 × 1 = 24.

The letters of the word 'MATH' can be arranged in 24 different ways.

40.
A 13-member club is choosing its President, Vice president, Secretary and Treasurer. Find the number of ways in which they can be chosen.
 a. 156 b. 1320 c. 1716 d. 17160

#### Solution:

President is to be chosen from 13 members, Vice president is to be chosen from 12 members, Secretary is to be chosen from 11 members and Treasurer is to be chosen from 10 members.

From the counting principle, the four people can be chosen in 13 x 12 x 11 x 10 ways.

= 17160
[Multiply.]

There are 17160 different ways in which they can be chosen.