All letters are used but first is a vowel?

Question

How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if 

all letters are used but first is a vowel?

Answer 1

Given : the word is ‘MONDAY.’ 

To find : possible number of words using all the letters of the word ‘MONDAY’ but the first letter is a vowel 

Total number of vowels = 2(A, O) 

Now, 

Fix the position of 1 vowel out of these two at first place which can be done in 2 ways.

A

or

O

Now, 

We need to fill the remaining 5 places 

Remaining places for letters = 5

So, arrange 5 letters at 5 places 

Formula used : 

Number of arrangements of n things taken all at a time = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\) 

∴ Total number of arrangements 

= the number of arrangements of 5 things taken all at a time 

= P(5, 5)

= \(\frac{5!}{(5-5)!}\)

= \(\frac{5!}{0!}\)

{∵ 0! = 1} 

= 5! 

= 5 × 4 × 3 × 2 × 1 

= 120 

Hence,

The total number of words can be made by using all letters of the word ‘MONDAY,’ but the first letter is vowel 

= 120 × 2 

= 240