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

#### Options

72

120

14

none of these.

#### Solution

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 B, H, R and T =\[4 \times 3 \times 2\]

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 B, H, R or 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.

∴ Total number of words that can be formed = 12 x 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.

∴ Total number of words having 2 A's = 4 x 3 =12

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