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Question
The number of arrangements of the letters of the word BHARAT taking 3 at a time is
Options
72
120
14
none of these.
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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
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