How many words, with or without meaning can be formed from the letters of the word 'MONDAY',

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 6C4 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 6C6 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 2C1 ways.

The remaining five letters can be chosen in 5C5 ways.

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

 

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