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How many words, with or without meaning can be formed from the letters of the word ‘MONDAY’, assuming that no letter is repeated, if
(i) 4 letters are used at a time

(ii) all letters are used at a time

(iii) all letters are used but first letter is a vowel ?

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Best answer

Given as

The word ‘MONDAY’

The total letters = 6

(i) 4 letters are used at a time

The number of ways = (No. of ways of choosing 4 letters from MONDAY)

= (6C4)

By using the formula,

nCr = n!/r!(n – r)!

6C4 = 6! / 4!(6 – 4)!

= 6! / (4! 2!)

= [6 × 5 × 4!] / (4! 2!)

= [6 × 5] / (2 × 1)

= 3 × 5

= 15

Then, we need to find the no. of words that can be formed by 4 letters.

15 × 4! = 15 × (4 × 3 × 2 × 1)

= 15 × 24

= 360

Hence, the no. of words that can be formed by 4 letters of MONDAY is 360.

(ii) all letters are used at a time

The total number of letters in the word ‘MONDAY’ is 6

Therefore, the total no. of words that can be formed is 6! = 360

Hence, the no. of words that can be formed by 6 letters of MONDAY is 360.

(iii) all letters are used but first letter is a vowel ?

In the word ‘MONDAY’ the vowels are O and A. We need to choose one vowel from these 2 vowels for the first place of the word.

Therefore,

The number of ways = (No. of ways of choosing a vowel from 2 vowels)

= (2C1)

By using the formula,

nCr = n!/r!(n – r)!

2C1 = 2! / 1!(2 – 1)!

= 2! / (1! 1!)

= (2 × 1)

= 2

Then we need to find the no. of words that can be formed by remaining 5 letters.

2 × 5! = 2 × (5 × 4 × 3 × 2 × 1)

= 2 × 120

= 240

Therefore, the no. of words that can be formed by all letters of MONDAY in which the first letter is a vowel is 240.

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