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

• A. 1580
• B. 1579
• C. 1582
• D. 1580

Hint:

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

Answer 1

The Correct Option is A

To determine the rank of the word 'UDAYPUR' when all its permutations are arranged in dictionary order, we follow these steps:

1. List the letters of 'UDAYPUR' in alphabetical order: A, D, P, R, U, U, Y.
2. Calculate the total number of permutations that can be formed with these letters, especially those that come before 'U'. Here, we consider permutations starting with letters before 'U'.
3. We first arrange permutations starting with letters A, D, P, and R, and then consider permutations starting with 'U' but where 'U' is followed by letters coming before 'D'. 
4. The key steps for finding the position of 'UDAYPUR' in dictionary order include:
5. Permutations starting with 'A': These can be calculated as \(\frac{6!}{2!}\) (since 'U' repeats).
6. Permutations starting with 'D': The calculation is similar \(\frac{6!}{2!}\).
7. Permutations starting with 'P': Calculate \(\frac{6!}{2!}\).
8. Permutations starting with 'R': Calculate \(\frac{6!}{2!}\).
9. Permutations starting with 'U', followed by a letter before 'D': No valid permutations.

For 'U' as the starting letter:

1. Permutations starting with 'UA': Calculate \(\frac{5!}{2!}\).
2. Permutations starting with 'UD': Consider permutations starting with 'UDA' (next lexicographically): This leads to \(\frac{4!}{1!}\).

The rank of permutations starting with 'UDP', where the words are 'UDAYP', 'UDAUP', ... etc., concludes the process before finding 'UDAYPUR'. Since the total of these numbers is 1579, the rank of 'UDAYPUR' specifically is 1579 + 1 = 1580.

Therefore, the rank of the word 'UDAYPUR' is 1580.