how many words combinations can you make with the name Rakesh Ramesh

Step 1: Identify all distinct letters and their frequencies

First, we list all the unique letters present in the combined phrase 'Rakesh Ramesh'. We also need to count the frequency of each letter, as repeated letters affect the number of unique combinations.

Step 2: Count the total number of letters

We sum up the frequency of all letters to find the total length of the sequence we need to arrange. In 'Rakesh Ramesh', we have the letters R, A, K, E, S, H, M. R appears twice, A twice, H twice, E twice, S once, K once, M once. Given 'Rakesh Ramesh', there are 12 letters in total.

Step 3: Apply the permutation formula for repeated letters

To find the number of unique word combinations (permutations) that can be made from a set of letters with repetitions, we use the formula for permutations with repetitions. Here, $n$ is the total number of letters, and $p_1, p_2, ..., p_k$ are the frequencies of each repeated letter.

Step 4: Substitute values into the formula

We substitute the total number of letters, $n=12$, and the frequencies of the repeated letters into the formula. The letter 'R' appears 2 times, 'A' appears 2 times, 'K' appears 1 time, 'E' appears 2 times, 'S' appears 1 time, 'H' appears 2 times, and 'M' appears 1 time. So the repeated frequencies are $2!, 2!, 2!, 2!$.

Step 5: Calculate the factorial values

We calculate the factorial of the total number of letters and the factorials of the frequencies of the repeated letters. $12!$ is $12 \times 11 \times ... \times 1$, and $2!$ is $2 \times 1$.

Step 6: Perform the division to find the total combinations

Finally, we divide $12!$ by the product of the factorials of the repeated letter frequencies. This gives us the total number of distinct word combinations that can be formed from the letters of 'Rakesh Ramesh'.

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