Find the number of permutations of the letters of the word ALLAHABAD.

Step 1: Count total letters and repeated letters

First, we count the total number of letters in the word ALLAHABAD. Then, we identify any letters that are repeated and count their occurrences. This is crucial for applying the permutation formula for words with repeated letters.

Step 2: Apply the permutation formula for repeated letters

The formula for permutations of $n$ objects where $p_1$ objects are of one type, $p_2$ objects are of a second type, and so on, up to $p_k$ objects of the $k$ -th type, is given by $n!$ divided by the product of the factorials of the counts of repeated letters.

Step 3: Substitute values into the formula

Now, we substitute the values we found into the formula. The total number of letters, $n$, is 9. The letter 'A' is repeated 4 times, so $p_1 = 4$. The letter 'L' is repeated 2 times, so $p_2 = 2$. The other letters (H, B, D) appear once, so their factorials are $1!$, which equals 1.

Step 4: Calculate the factorials

Next, we calculate the factorial values for each number. $9!$ means $9 \times 8 \times \dots \times 1$, $4!$ means $4 \times 3 \times 2 \times 1$, and $2!$ means $2 \times 1$.

Step 5: Compute the final number of permutations

Finally, we substitute the calculated factorial values back into the formula and perform the division to find the total number of distinct permutations.

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