How many of these in word "BHARAT" begin with B and end with T ?

Answer

To find how many arrangements of the letters in the word "BHARAT" begin with 'B' and end with 'T', we can follow these steps:

### Step 1: Identify the fixed letters
The word "BHARAT" has 6 letters: B, H, A, R, A, T. Since we want arrangements that begin with 'B' and end with 'T', we fix these letters in their positions:
- First letter: B
- Last letter: T

### Step 2: Determine the letters to arrange
After fixing 'B' at the start and 'T' at the end, we are left with the letters in the middle: H, A, R, A.

### Step 3: Count the letters in the middle
The letters we need to arrange in the middle are H, A, R, A. This gives us a total of 4 letters, but note that the letter 'A' is repeated.

### Step 4: Calculate the arrangements
The formula for the number of arrangements of n items where there are repetitions is given by:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
where \( n \) is the total number of items, and \( p_1, p_2, \ldots \) are the frequencies of the repeated items.

In our case:
- Total letters (n) = 4 (H, A, R, A)
- The letter 'A' is repeated 2 times.

Thus, the number of arrangements is:
\[ \text{Number of arrangements} = \frac{4!}{2!} \]

### Step 5: Calculate factorials
Now we calculate the factorials:
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Final calculation
Now, substituting back into the formula:
\[ \text{Number of arrangements} = \frac{24}{2} = 12 \]

### Conclusion
Thus, the total number of arrangements of the letters in "BHARAT" that begin with 'B' and end with 'T' is **12**.

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