Given:

The word ‘MONDAY’

Total letters = 6

**(i) **4 letters are used at a time

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

= (^{6}C_{4})

By using the formula,

^{n}C_{r} =\( \dfrac{ n!}{r!(n – r)!}\)

^{6}C_{4} = \(\dfrac{ 6! }{ 4!(6 – 4)!}\)

= 3×5

= 15

Now 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

The no. of words that can be formed by 4 letters of MONDAY is 360.

**(ii) **all letters are used at a time

Total number of letters in the word ‘MONDAY’ is 6

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

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.

So,

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

= (^{2}C_{1})

By using the formula,

^{n}C_{r} =\( \dfrac{ n!}{r!(n – r)!}\)

^{2}C_{1} = \( \dfrac{2! }{ 1!(2 – 1)!}\)

= (2×1)

= 2

Now 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

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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