The number of a arrangement of the letter of the word BHARAT taking 3 at a time is

Options

A. 72
B. 120
C. 14
D. none of these 0

Answer

To find the number of arrangements of the letters of the word "BHARAT" taken 3 at a time, we can break down the problem into two cases based on the letters we can choose.

### Step 1: Identify the letters and their frequencies
The word "BHARAT" consists of the letters: B, H, A, R, A, T.
- B: 1 time
- H: 1 time
- A: 2 times
- R: 1 time
- T: 1 time

### Step 2: Case 1 - All three letters are different
In this case, we can choose 3 different letters from the available 5 unique letters (B, H, A, R, T).

- The number of ways to choose 3 letters from 5 is given by the combination formula:
\[ \text{Number of ways} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

- Each selection of 3 letters can be arranged in \(3!\) (factorial of 3) ways:
\[ 3! = 6 \]

- Therefore, the total arrangements for this case is:
\[ \text{Total arrangements} = \binom{5}{3} \times 3! = 10 \times 6 = 60 \]

### Step 3: Case 2 - Two letters are the same and one is different
In this case, we can only use the letter 'A' twice (since it is the only letter that appears more than once). We need to choose one different letter from the remaining letters (B, H, R, T).

- The number of ways to choose 1 letter from the remaining 4 letters is:
\[ \text{Number of ways} = \binom{4}{1} = 4 \]

- The arrangement of the letters A, A, and the chosen different letter can be calculated as follows (since we have two A's, we divide by \(2!\)):
\[ \text{Arrangements} = \frac{3!}{2!} = \frac{6}{2} = 3 \]

- Therefore, the total arrangements for this case is:
\[ \text{Total arrangements} = \binom{4}{1} \times \frac{3!}{2!} = 4 \times 3 = 12 \]

### Step 4: Combine the results from both cases
Finally, we add the total arrangements from both cases to get the overall total:
\[ \text{Total arrangements} = 60 + 12 = 72 \]

### Final Answer
The total number of arrangements of the letters of the word "BHARAT" taken 3 at a time is **72**.

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