Question:

**Step 1:** Count the total number of letters in the word ALLAHABAD. The word ALLAHABAD contains $9$ letters. **Step 2:** Identify the frequency of each distinct letter. * 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 with repetition. The number of distinct words that can be formed using all the letters of a given word is calculated by the formula: $$ \frac{n!}{n_1! n_2! \cdots n_k!} $$ where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the frequencies of the distinct letters. Substituting the values for ALLAHABAD: $$ \text{Number of words} = \frac{9!}{4! \times 2! \times 1! \times 1! \times 1!} $$ **Step 4:** Calculate the number of words. $$ \text{Number of words} = \frac{362,880}{24 \times 2 \times 1 \times 1 \times 1} $$ $$ \text{Number of words} = \frac{362,880}{48} $$ $$ \text{Number of words} = 7,560 $$ Thus, $7,560$ distinct words can be formed using all the letters of the word ALLAHABAD.