Permutations and Combinations Worksheet

**Page 3**

21.

In how many ways can 6 people sit in 6 chairs?

a. | 721 | ||

b. | 5040 | ||

c. | 120 | ||

d. | 720 |

= 6! = 6 ×5 ×4 ×3 ×2 ×1 = 720

[Simplify.]

6 people can sit in 6 chairs in 720 ways.

Correct answer : (4)

22.

Find: ($\frac{\mathrm{5!}}{\mathrm{3!}}$ ) × ($\frac{1}{\mathrm{2!}}$ )

a. | 15 | ||

b. | 10 | ||

c. | 5 | ||

d. | None of the above |

[Expand 5!.]

3! = 3 ×2 ×1

[Expand 3!.]

2! = 2 ×1 = 2

[Expand 2! and multiply.]

(

[Simplify the fraction.]

(

[Substitute the value of 2!.]

(

= 10

[Simplify.]

The value of (

Correct answer : (2)

23.

Paul lists different arrangements of the letters in the word $\mathrm{JON}$ as follows: $\mathrm{JNO}$, $\mathrm{JON}$, $\mathrm{NOJ}$, $\mathrm{NJO}$, and $\mathrm{ONJ}$. Find the number of arrangements not included in the list.

a. | 2 | ||

b. | 1 | ||

c. | 3 | ||

d. | 8 |

= 3! = 3 x 2 = 6

[Expand 3! and multiply.]

Total number of arrangements formed by all the letters of the word

Total number of arrangements listed = 5

Number of arrangements not included in the list = 6 - 5 = 1

[Subtract.]

Correct answer : (2)

24.

In how many ways can 9 questions be selected out of 14 questions such that the selection always contains the questions numbered 1, 2 and 7?

a. | 452 | ||

b. | 462 | ||

c. | 472 | ||

d. | 467 |

Number of questions remaining after selecting questions numbered 1, 2 and 7 = 14 - 3 = 11

The remaining 6 questions have to be selected from these 11 questions.

Number of ways in which 6 questions can be selected out of 11 questions =

6! = 6 ×5 ×4 ×3 ×2 ×1 = 720

[Expand 6! and multiply.]

[Expand

[Substitute the values of 6! and

The number of ways in which 9 questions can be selected out of 14 questions, always selecting questions numbered 1, 2 and 7 is 462.

Correct answer : (2)

25.

Which tree diagram represents the possible combinations of the spins of the spinners 1 and 2?

a. | Figure A | ||

b. | Figure B |

The possible outcomes of spinner 2 are 1 or 2. So, the number of outcomes of spinner 2 = 2

For every outcome of spinner 1 there should be a outcome of spinner 2.

So, the number of possible combinations of the spins of the spinners 1 and 2 are (A, 1), (A, 2), (B, 1),(B, 2), (C, 1), (C, 2), (D, 1), and (D, 2). These are represented as a tree diagram as shown below.

The above tree diagram matches with the tree diagram in figure B.

Correct answer : (2)

26.

Stephanie has 7 books to arrange in a shelf. In how many ways can she arrange them?

a. | 720 | ||

b. | 120 | ||

c. | 5040 | ||

d. | None of the above |

There are 7 books to fill the first place in the shelf, 6 books to fill the second place, 5 books to fill the third place in the shelf and so on.

By counting principle,

1

7 × 6 × 5 . . . 1 = 5040

So, there are 5040 different ways in which Stephanie can arrange 7 books.

Correct answer : (3)

27.

Find the number of combinations of choosing 2 cubes from 5 cubes shown.

a. | 10 | ||

b. | 72 | ||

c. | 56 | ||

d. | 25 |

The list of all the possible arrangements of 2 cubes is:

(Red, Yellow), (Red, Blue), (Red, Green), (Red, White),

(Yellow, Red), (Yellow, Blue), (Yellow, Green), (Yellow, White),

(Blue, Red), (Blue, Yellow), (Blue, Green), (Blue, White),

(Green, Red), (Green, Yellow), (Green, Blue), (Green, White)

(White, Red), (White, Yellow), (White, Blue), (White, Green).

As choosing (Red, Yellow) is same as (Yellow, Red), cancel the duplicate arrangements as shown below.

The list of all the combinations of 2 cubes after canceling the duplicates is:

(Red, Yellow), (Red, Blue), (Red, Green), (Red, White),

(Yellow, Blue), (Yellow, Green), (Yellow, White),

(Blue, Green), (Blue, White),

(Green, White)

So, the number of combinations of choosing 2 cubes from 5 cubes is 10.

Correct answer : (1)

28.

A school has 14 teachers out of which 8 are male. A team of 8 teachers is to be formed with 4 female teachers. In how many ways can the team be formed?

a. | 1060 | ||

b. | 1050 | ||

c. | 1070 | ||

d. | 1040 |

[Subtract.]

Number of ways to select 4 female teachers out of 6 female teachers =

Number of ways to select 4 male teachers out of 8 male teachers =

Number of ways in which the team can be formed = Number of ways to select 4 female teachers × Number of ways to select 4 male teachers

= 15 × 70 = 1050

[Substitute and multiply.]

A team of 8 teachers can be formed in 1050 different ways.

Correct answer : (2)

29.

Find the value of $r$, if _{7}P_{4} + $r$ x _{7}C_{4} = 0.

a. | -24 | ||

b. | 24 | ||

c. | -27 | ||

d. | 21 |

[Original equation.]

[Expand

840 +

[Substitute

840 +

[Cancel 840 with 24.]

[Subtract -840 from each side.]

[Divide each side by 35.]

The value of

Correct answer : (1)

30.

A number lock has 9 different digits. A combination of two different digits can be set to open the lock. How many combinations are possible?

a. | 81 | ||

b. | 72 | ||

c. | 63 | ||

d. | 18 |

Number of digits remaining after selecting the first number = 9 - 1 = 8

Number of ways of selecting the second number out of 8 numbers =

Total number of possible combinations = Number of ways of selecting the first number × Number of ways of selecting the second number = 9 × 8 = 72

72 combinations can be set to open the lock.

Correct answer : (2)