How many different words can be formed by using all the letters of the word 'ALLAHABAD'?

Explanation

To find the number of different words that can be formed using all the letters of the word "ALLAHABAD", we will 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 "ALLAHABAD", the frequency of each letter is as follows:
- 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 to calculate the number of distinct permutations of a word with repeated letters is given by:

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

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 9 \) (total letters)
- \( n_A = 4 \) (for letter A)
- \( n_L = 2 \) (for letter L)
- \( n_H = 1 \) (for letter H)
- \( n_B = 1 \) (for letter B)
- \( n_D = 1 \) (for letter D)

### Step 4: Substitute the values into the formula
Now we can substitute the values into the formula:

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

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

### Step 6: Substitute the factorial values into the formula
Substituting the calculated values:

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

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

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

### Final Answer
Thus, the total number of different words that can be formed using all the letters of the word "ALLAHABAD" is **7560**.

---