Question

The number of arrangements of the letters of the word BHARAT taking 3 at a time is

• A. $72$
• B. $120$
• C. $14$
• D. none of these

Answer 1

The correct option is A $72$

Given, word $BHARAT$

There are 2 cases

(i) the one including no repeating letters i.e. only 1 A
i.e we have to take 3 letters out of the five letters $B,H,A,R,T$
There are ${}^5{P}_3=60$ ways to choose 3 out of 5 and arrange them

(ii) the words with repeated letters i.e 2A's

then you have to choose one from the 4 letters $B,H,R,T$ and $2A^{\prime }s$ which must be selected to form a word of length 3

Then you have $(4\times \frac{3!}{2!}$) different arrangements of 3 letters, containing double A

Makes the total $60+12=72$

Hence Option $A$ is correct