**Question:**

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

How many words, with or without meaning can be made 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 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of 6 different objects taken 4 at a time, which is ${ }^{6} \mathrm{P}_{4}$.

Thus, required number of words that can be formed using 4 letters at a time is

${ }^{6} P_{4}=\frac{6 !}{(6-4) !}=\frac{6 !}{2 !}=\frac{6 \times 5 \times 4 \times 3 \times 2 !}{2 !}=6 \times 5 \times 4 \times 3=360$

(ii) Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations of 6 different objects taken 6 at a time, ${ }^{6} \mathrm{P}_{6}=6 !$

Thus, required number of words that can be formed when all letters are used at a time = 6! = 6 × 5 × 4 × 3 × 2 ×1 = 720

(iii) In the given word, there are 2 different vowels, which have to occupy the rightmost place of the words formed. This can be done only in 2 ways.

Since the letters cannot be repeated and the rightmost place is already occupied with a letter (which is a vowel), the remaining five places are to be filled by the remaining 5 letters. This can be done in 5! ways.

Thus, in this case, required number of words that can be formed is

5! × 2 = 120 × 2 = 240