Question

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?

Answer 1

There are six letters in the word MONDAY.

(i) 4 letters are used at a time:
Four letters can be chosen out of six letters in ⁶C₄ ways.
So, there are ⁶C₄ groups containing four letters that can be arranged in $4!$ ways.
∴ Number of ways =${}^6{C}_4\times 4!=\frac{6!}{4!2!}\times 4!=\frac{6!}{2!}=360$

(ii) All the letters are used at a time:
This can be done in ⁶C₆ ways.
So, there are ⁶C₆ groups containing six letters that can be arranged in $6!$ ways.
∴ Number of ways =${}^6{C}_6\times 6!=1\times 720=720$

(iii) All the letters are used, but the first letter is a vowel:
There are two vowels, namely A and O, in the word MONDAY.
For the first letter, out of the two vowels, one vowel can be chosen in ²C₁ ways.
The remaining five letters can be chosen in ⁵C₅ ways.
So, the letters in ⁵C₅ group can be arranged in $5!$ ways.
∴ Number of ways =${}^2{C}_1{\times }^5{C}_5\times 5!=2\times 1\times 5!=240$