Find the number of permutations of the letter of the words
'ALLAHABAD '

Answer

To find the number of permutations of the letters in the word "ALLAHABAD", we will follow these steps:

### Step 1: Count the total number of letters
The word "ALLAHABAD" consists of 9 letters.

### Step 2: Identify the repeating letters
In the word "ALLAHABAD":
- The letter 'A' appears 4 times.
- The letter 'L' appears 2 times.
- The letter 'H' appears 1 time.
- The letter 'B' appears 1 time.
- The letter 'D' appears 1 time.

### Step 3: Apply the formula for permutations of multiset
The formula for the number of permutations of a multiset is given by:

\[ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where:
- \( n \) is the total number of items (letters in this case),
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating items.

For "ALLAHABAD":
- \( n = 9 \) (total letters)
- \( p_1 = 4 \) (for 'A')
- \( p_2 = 2 \) (for 'L')
- \( p_3 = 1 \) (for 'H')
- \( p_4 = 1 \) (for 'B')
- \( p_5 = 1 \) (for 'D')

### Step 4: Substitute the values into the formula
Using the values we have:

\[ \text{Number of permutations} = \frac{9!}{4! \times 2! \times 1! \times 1! \times 1!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:

- \( 9! = 362880 \)
- \( 4! = 24 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

### Step 6: Substitute the factorial values into the equation
Now substituting these values into the formula:

\[ \text{Number of permutations} = \frac{362880}{24 \times 2 \times 1 \times 1 \times 1} = \frac{362880}{48} \]

### Step 7: Perform the division
Now, divide \( 362880 \) by \( 48 \):

\[ \text{Number of permutations} = 7560 \]

### Final Answer
Thus, the number of permutations of the letters of the word "ALLAHABAD" is **7560**.

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