**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?

**Solution:**

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 6*C*4 ways.

So, there are ${ }^{6} \mathrm{C}_{4}$ groups containing four letters that can be arranged in $4 !$ ways.

$\therefore$ 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 6*C*6 ways.

So, there are ${ }^{6} C_{6}$ groups containing six letters that can be arranged in $6 !$ ways.

$\therefore$ 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 2*C*1 ways.

The remaining five letters can be chosen in 5*C*5 ways.

So, the letters in ${ }^{5} C_{5}$ group can be arranged in $5 !$ ways.

$\therefore$ Number of ways $={ }^{2} C_{1} \times{ }^{5} C_{5} \times 5 !=2 \times 1 \times 5 !=240$