Permutations and Combinations Worksheet

**Page 10**

91.

What is the value of$\frac{5!}{(5-2)!}$?

a. | 20 | ||

b. | 40 | ||

c. | 25 | ||

d. | 23 |

[Expand 5!.]

(5 - 2)! = 3!

[Simplify.]

[Substitute the values of 5! and (5 - 2)!.]

= 5 × 4 = 20

[Simplify.]

The value of

Correct answer : (1)

92.

What is the value of $\frac{\mathrm{5!}}{\mathrm{(5-2)!}}$?

a. | 25 | ||

b. | 20 | ||

c. | 40 | ||

d. | 23 |

[Expand 5!.]

(5 - 2)! = 3!

[Simplify.]

[Substitute the values of 5! and (5 - 2)!.]

= 5 x 4 = 20

[Simplify.]

The value of

Correct answer : (2)

93.

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

a. | 720 | ||

b. | 320 | ||

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.]

1st letter 2nd letter 3rd letter 4th letter 5th letter 6th letter

6 x 5 x 4 x 3 x 2 x 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 : (1)

94.

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

a. | 520 | ||

b. | 420 | ||

c. | 720 | ||

d. | 320 |

[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 | x | 5 | x | 4 | x | 3 | x | 2 | x | 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 : (3)

95.

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

a. | 8 | ||

b. | 3 | ||

c. | 10 | ||

d. | 6 |

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 : (1)

96.

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

a. | 3 | ||

b. | 10 | ||

c. | 8 | ||

d. | 6 |

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 : (3)

97.

Find the number of ways of forming a password with 4 letters and 4 digits to create a mail id (repetition not allowed).

a. | _{26}C_{4} x _{10}C_{4} x 8! | ||

b. | _{26}C_{4} x _{10}C_{4} | ||

c. | _{26}P_{4} x _{10}P_{4} x 8! | ||

d. | _{36}C_{8} |

Two digits out of 10 digits can be chosen in

The number of ways of arranging each of the selections = 8!

So, the total number of ways of forming the password =

Correct answer : (-1)

98.

In how many ways can you choose 2 even numbers and 3 odd numbers from the numbers from 3, 4, 5, 6, 7, 8 and 9?

a. | 12 | ||

b. | 7 | ||

c. | 10 | ||

d. | 14 |

3, 5, 7 and 9 are the odd numbers, which are 4 in number.

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

(4, 6), (4, 8)

(6, 8)

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

(3, 5, 7), (3, 5, 9)

(5, 7, 9), (3, 7, 9)

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

(4, 6, 3, 5, 7), (4, 6, 3, 5, 9), (4, 6, 5, 7, 9), (4, 6, 3, 7, 9)

(4, 8, 3, 5, 7), (4, 8, 3, 5, 9), (4, 8, 5, 7, 9), (4, 8, 3, 7, 9)

(6, 8, 3, 5, 7), (6, 8, 3, 5, 9), (6, 8, 5, 7, 9), (6, 8, 3, 7, 9)

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

Correct answer : (1)

99.

William lists different arrangements of the letters in the word $\mathrm{NAD}$ as follows: $\mathrm{NDA}$, $\mathrm{NAD}$, $\mathrm{DAN}$, $\mathrm{DNA}$, and $\mathrm{ADN}$. Find the number of arrangements not included in the list.

a. | 3 | ||

b. | 2 | ||

c. | 1 | ||

d. | 8 |

= 3! = 3 x 2 x 1 = 6

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

[Expand 3! and multiply]

Total number of arrangements listed = 5

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

[Subtract.]

Correct answer : (3)

100.

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

a. | 12 | ||

b. | 6 | ||

c. | 9 | ||

d. | 3 |

FCP, FPC, CPF, CFP, PFC and PCF

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

Correct answer : (2)