Question - Determining the Dictionary Rank of a Specific Arrangement of Letters

Let's calculate the rank of the word RACHIT.

Step 1: Write down the letters in alphabetical order: A, C, H, I, R, T.

Step 2: Fix the first letter and calculate the number of permutations for the remaining letters.

We have 5 positions left, so we will calculate \(5!\) permutations for the following scenarios:

• R is fixed (A, C, H, I, T): count the permutations starting with A, C, H, and I.

\(4 \times 5! = 4 \times 120 = 480\)

Step 3: Now fix the second letter and calculate permutations for the remaining letters, until the selected letter exceeds the corresponding letter in RACHIT.

For the word RACHIT, after R, A is fixed:

• R, A is fixed (C, H, I, T): count the permutations starting with C and H.

\(2 \times 4! = 2 \times 24 = 48\)

Step 4: Repeat step 3 for all subsequent letters.

• R, A, C is fixed (H, I, T): count the permutations starting with H.

\(1 \times 3! = 1 \times 6 = 6\)

Finally, add all these permutations:

480 (for R) + 48 (for RA) + 6 (for RAC) + 1 (for the next turn RACHI to RACHIT)

Total rank = 480 + 48 + 6 + 1 = 535

So the rank of the word RACHIT is 535.