how many word combinations can be made with the names in Vijayawada

Step 1: Count total letters

First, we count the total number of letters in the word VIJAYAWADA. There are 10 letters in total.

Step 2: Identify repeated letters and their counts

Next, we identify any letters that are repeated and count how many times each repeated letter appears. In the word VIJAYAWADA, only the letter 'A' is repeated.

Step 3: Apply permutation formula for repeated letters

To find the number of distinct word combinations (permutations) that can be made from the letters of the word, we use the formula for permutations with repeated items. Here, $n$ is the total number of letters and $p_1, p_2, ..., p_k$ are the frequencies of the repeated letters. Since only 'A' is repeated 4 times, we divide $10!$ by $4!$.

Step 4: Calculate the factorial values

Now we calculate the factorial values for $10!$ and $4!$. This involves multiplying all positive integers up to each number.

Step 5: Compute the final number of permutations

Finally, we divide the total number of permutations (if all letters were distinct) by the factorial of the count of each repeated letter. This gives us the number of unique word combinations that can be formed.

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