**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.

**Solution:**

(a) 72

When we make words after selecting letters of the word BHARAT, it could consist of a single A, two As or no A.

Case-I: A is not selected for the three letter word.

Number of arrangements of three letters out of $\mathrm{B}, \mathrm{H}, \mathrm{R}$ and $\mathrm{T}=4 \times 3 \times 2=24$

Case-II: One A is selected and the other two letters are selected out of B, H, R or T.

Possible ways of selection: Selecting two letters out of $\mathrm{B}, \mathrm{H}, \mathrm{R}$ or $\mathrm{T}$ can be done in ${ }^{4} P_{2}=12$ ways.

Now, in each of these 12 ways, these two letters can be placed at any of the three places in the three letter word in 3 ways.

$\therefore$ Total number of words that can be formed $=12 \times 3=36$

Case-III: Two A's and a letter from B, H, R or T are selected.

Possible ways of arrangement:

Number of ways of selecting a letter from B, H, R or T = 4

And now this letter can be placed in any one of the three places in the three letter word other than the two A's in 3 ways.

$\therefore$ Total number of words having $2 \mathrm{~A}^{\prime} \mathrm{s}=4 \times 3=12$

Hence, total number of words that can be formed $=24+36+12=72$