In how many different ways can the letters of the word 'ALLAHABAD' be arranged?

Options

A. 7560
B. 7840
C. 7460
D. 7650

Answer

To find the number of different ways the letters of the word "ALLAHABAD" can be arranged, we can follow these steps:

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

### Step 2: Identify the frequency of each letter
In the word "ALLAHABAD", the letters and their frequencies are:
- A: 4 times
- L: 2 times
- H: 1 time
- B: 1 time
- D: 1 time

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

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]

where:
- \( n \) is the total number of items,
- \( n_1, n_2, n_3, \ldots \) are the frequencies of the distinct items.

### Step 4: Apply the formula
Here, \( n = 9 \) (total letters), \( n_1 = 4 \) (for A), \( n_2 = 2 \) (for L), \( n_3 = 1 \) (for H), \( n_4 = 1 \) (for B), and \( n_5 = 1 \) (for D).

So, we can substitute these values into the formula:

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

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 9! = 362880 \)
- \( 4! = 24 \)
- \( 2! = 2 \)
- \( 1! = 1 \) (for H, B, and D)

### Step 6: Substitute and simplify
Now we substitute these values into the equation:

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

### Step 7: Perform the division
Now, we perform the division:

\[ \frac{362880}{48} = 7560 \]

### Final Answer
Thus, the number of different ways the letters of the word "ALLAHABAD" can be arranged is **7560**.

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