Question - Ranking Permutations of a Given Word

The word RACHIT contains 6 distinct letters.

To find the rank of the word RACHIT when the letters are arranged in dictionary order:

Step 1: Arrange the letters of the word in alphabetical order: A, C, H, I, R, T.

Step 2: Count the number of words starting with each letter that is before R alphabetically and with the rest in any order.

Step 3: For A as the first letter, we can arrange the remaining 5 letters in \(5!\) ways.

Step 4: For C as the first letter, the count is again \(5!\) ways.

Step 5: For H as the first letter, the count is \(5!\) ways.

Step 6: For I as the first letter, the count is \(5!\) ways.

Step 7: With R as the first letter, the next letter could be A, which gives \(4!\) arrangements for the remaining letters.

Therefore, the rank of RACHIT = the sum of all the arrangements calculated above

\(= 4 \times 5! + 1 \times 4!\)

\(= 4 \times (5 \times 4 \times 3 \times 2 \times 1) + 4 \times 3 \times 2 \times 1\)

\(= 4 \times 120 + 24\)

\(= 480 + 24\)

\(= 504\)

So, the rank of the word RACHIT is 504 when the letters are arranged in dictionary order.