Permutations and Combinations Worksheet

**Page 7**

61.

In how many ways can you choose 2 even numbers and 3 odd numbers from the numbers 1, 2, 3, 4, 5, 6, 7?

a. | 36 | ||

b. | 28 | ||

c. | 14 | ||

d. | 12 |

1, 3, 5 and 7 are the odd numbers, which are 4 in number.

List all the possible combinations of choosing 2 even numbers as follows:

(2, 4), (2, 6)

(4, 6)

List all the possible combinations of choosing 3 odd numbers as follows:

(1, 3, 5), (1, 3, 7)

(3, 5, 7), (1, 5, 7)

List all the possible combinations of choosing 2 even numbers and 3 odd numbers as follows:

(2, 4, 1, 3, 5), (2, 4, 1, 3, 7), (2, 4, 3, 5, 7), (2, 4, 1, 5, 7)

(2, 6, 1, 3, 5), (2, 6, 1, 3, 7), (2, 6, 3, 5, 7), (2, 6, 1, 5, 7)

(4, 6, 1, 3, 5), (4, 6, 1, 3, 7), (4, 6, 3, 5, 7), (4, 6, 1, 5, 7)

So, 2 even numbers and 3 odd numbers can be chosen in 12 different ways.

Correct answer : (4)

62.

In how many ways can you arrange the letters in the word RANDOM?

a. | 320 | ||

b. | 720 | ||

c. | 420 | ||

d. | 520 |

[There are 6 letters in the word.]

Number of possible outcomes for choosing the second letter = 5

[There are 5 letters remaining to choose.]

Number of possible outcomes for choosing the third letter = 4

[There are 4 letters remaining to choose after choosing 2 letters.]

Number of possible outcomes for choosing the fourth letter = 3

[There are 3 letters remaining to choose after choosing 3 letters.]

Number of possible outcomes for choosing the fifth letter = 2

[There are 2 letters remaining to choose after choosing 4 letters.]

Number of possible outcomes for choosing the sixth letter = 1

[There is only 1 letter remaining to choose after choosing 5 letters.]

1^{st}letter | 2^{nd}letter | 3^{rd}letter | 4^{th}letter | 5^{th}letter | 6^{th}letter | |||||||

6 | × | 5 | × | 4 | × | 3 | × | 2 | × | 1 | = | 720 |

[Multiply the number of possible outcomes of choosing each letter in the word.]

The letters in the word can be arranged in 720 ways.

Correct answer : (2)

63.

In how many ways can 8 employees be selected from 80 employees to form a committee?

a. | $\frac{{}_{80}{\mathrm{P}}_{8}}{8!}$ | ||

b. | $\frac{{}_{80}{\mathrm{P}}_{8}}{80!}$ | ||

c. | _{80}P_{8} | ||

d. | None of the above |

Number of employees to be selected = 8

Number of ways to select 8 employees from 80 employees =

There are

Correct answer : (1)

64.

Find the number of three letter words that can be formed with the letters of the word TOP.

a. | 9 | ||

b. | 6 | ||

c. | 3 | ||

d. | 12 |

TOP, TPO, OPT, OTP, PTO and POT

So, the number of words that can be formed with the letters of the word TOP is 6.

Correct answer : (2)

65.

John went to a restaurant. The menu chart contains a list of 7 dishes. He wanted to order 5 dishes from the menu. State whether the way of ordering the dishes is a permutation or a combination.

a. | Combination | ||

b. | Permutation |

The sequence of ordering dishes is not important here and different sequence of orders of same set of dishes form the same combination.

So, the ordering of dishes is a combination.

Correct answer : (1)

66.

Victor has to answer 2 questions out of 4 questions in an examination. State whether his selection of 2 questions is a permutation or a combination.

a. | Combination | ||

b. | Permutation |

The sequence of selecting 2 questions is not important here and different sequence of orders of same set of questions form the same selection.

So, the selection of questions is a combination.

Correct answer : (1)

67.

State whether the arrangement of 3-lettered words from the alphabets C, H, E, M, I, S, T, R, Y is a permutation or a combination.

a. | Permutation | ||

b. | Combination |

The ordering of alphabets is important here and different orders of same letters form different words.

So, the arrangement of 3-lettered words from the alphabets is a permutation.

Correct answer : (1)

68.

What is the value of _{3}P_{6}?

a. | 3 | ||

b. | 6 | ||

c. | 18 | ||

d. | Not defined |

The value of

In

The value of

Correct answer : (4)

69.

How many outcomes are possible, when three coins are tossed?

a. | 6 | ||

b. | 10 | ||

c. | 3 | ||

d. | 8 |

Each coin has the outcome of either a Head (H) or Tail (T).

The possible outcomes of tossing three coins is HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

There are 8 possible outcomes when the three coins are tossed.

Correct answer : (4)

70.

John plans to study, watch TV and play football on a Sunday. In how many ways can he choose and arrange the activities?

a. | 2 | ||

b. | 6 | ||

c. | 3 | ||

d. | 1 |

First activity can be chosen in 3 ways; second activity can be chosen in 2 ways and third activity can be chosen in 1 way.

By counting principle,

1

3 × 2 × 1 = 6

John can choose and arrange the 3 activities in 6 different ways.

Correct answer : (2)