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.