Question

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

Hint:

While finding dictionary rank, always arrange letters alphabetically first and divide by factorials of repeated letters.

Answer 1

Correct Answer: 1580

Step 1: Arrange the letters alphabetically.
The letters of the word UDAYPUR are U, D, A, Y, P, U, R.
Arranging them in alphabetical order gives: A, D, P, R, U, U, Y.
Step 2: Count permutations starting with letters before U.
Letters before U are A, D, P, and R.
For each such letter, the remaining letters can be arranged in:
\[ \frac{6!}{2!} = 360 \] Total permutations before U:
\[ 4 \times 360 = 1440 \] Step 3: Fix U as the first letter and proceed sequentially.
For second letter A:
\[ \frac{5!}{2!} = 120 \] For third letter D:
\[ 3! = 6 \] For fourth letter A:
\[ 3! = 6 \] For fifth letter Y:
\[ 3! = 6 \] For sixth letter P:
\[ 1 \] For seventh letter R:
\[ 1 \] Step 4: Add all permutations.
\[ 1440 + 120 + 6 + 6 + 6 + 1 + 1 = 1580 \] Step 5: Final conclusion.
Hence, the rank of the word UDAYPUR is 1580.