Permutations and Combinations

What are the number of ways to select 3 men and 2 women such that one man and one woman are always selected?

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What are the number of ways to select 3 men and 2 women such that one man and one woman are always selected?

A. 100
B. 60
C. 30
D. 20
Explanation:
The number of ways to select three men and two women such that one man and one woman are always selected = Number of ways selecting two men and one woman from men and five women
= ⁴C₂ * ⁵C₁ = (4 * 3)/(2 * 1) * 5
= 30 ways.

The number of permutations of the letters of the word ‘MESMERISE’ is___________?

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The number of permutations of the letters of the word ‘MESMERISE’ is___________?

A. 9!/(2!)2 3!
B. 9!/(2!)3 3!
C. 9!/(2!)2 (3!)2
D. 5!/(2!)2 3!
Explanation:
n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.
The letter pattern ‘MESMERISE’ consists of 10 letters of which there are 2M’s, 3E’s, 2S’s and 1I and 1R.
Number of arrangements = 9!/(2!)2 3!

The number of arrangements that can be made with the letters of the word MEADOWS so that the vowels occupy the even places?

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The number of arrangements that can be made with the letters of the word MEADOWS so that the vowels occupy the even places?

A. 720
B. 144
C. 120
D. 36
Explanation:
The word MEADOWS has 7 letters of which 3 are vowels.
-V-V-V-
As the vowels have to occupy even places, they can be arranged in the 3 even places in 3! i.e., 6 ways. While the consonants can be arranged among themselves in the remaining 4 places in 4! i.e., 24 ways.
Hence the total ways are 24 * 6 = 144.

Using all the letters of the word “NOKIA”, how many words can be formed, which begin with N and end with A?

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Using all the letters of the word “NOKIA”, how many words can be formed, which begin with N and end with A?

A. 3
B. 6
C. 24
D. 120
Explanation:
There are five letters in the given word.
Consider 5 blanks ….
The first blank and last blank must be filled with N and A all the remaining three blanks can be filled with the remaining 3 letters in 3! ways.
The number of words = 3! = 6.

A boy has nine trousers and 12 shirts. In how many different ways can he select a trouser and a shirt?

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A boy has nine trousers and 12 shirts. In how many different ways can he select a trouser and a shirt?

A. 21
B. 12
C. 9
D. 108

Explanation:
The boy can select one trouser in nine ways.
The boy can select one shirt in 12 ways.
The number of ways in which he can select one trouser and one shirt is 9 * 12 = 108 ways.