Rank Calculation for UDAYPUR Word

To find the rank of the word "UDAYPUR" when arranged in a dictionary, we follow these steps:

UDAYPUR Word Letter Analysis

• The word is "UDAYPUR".
• It has 7 letters: U, D, A, Y, P, U, R.
• The letters are: A, D, P, R, U, U, Y.
• There are 7 letters in total, with the letter 'U' repeated 2 times.
• The distinct letters in alphabetical order are: A, D, P, R, U, Y.

Counting Words Before UDAYPUR

We count the number of words that come before "UDAYPUR" in alphabetical order.

Permutations Starting Before 'U'

• Letters smaller than 'U' in the set {A, D, P, R, U, Y} are A, D, P, R.
• For each of these 4 letters, we fix it as the first letter and find the permutations of the remaining 6 letters. Since 'U' is repeated twice, the formula is $ \frac{6!}{2!} $.
• Number of permutations for each starting letter = $ \frac{720}{2} = 360 $.
• Total words starting with A, D, P, R = $ 4 \times 360 = 1440 $.

Permutations Starting 'U' then Before 'D'

• The word starts with 'U'. Remaining letters: {A, D, P, R, U, Y}.
• The second letter is 'D'. Letters smaller than 'D' in the remaining set is 'A'.
• We fix 'UA'. Remaining letters: {D, P, R, U, Y} (5 distinct letters).
• Number of permutations = $ 5! = 120 $.
• Total words starting with 'UA' = $ 1 \times 120 = 120 $.

Permutations Starting 'UD' then Before 'A'

• The word starts with 'UD'. Remaining letters: {A, P, R, U, Y}.
• The third letter is 'A'. There are no letters smaller than 'A' in the remaining set.
• Total words = $ 0 $.

Permutations Starting 'UDA' then Before 'Y'

• The word starts with 'UDA'. Remaining letters: {P, R, U, Y}.
• The fourth letter is 'Y'. Letters smaller than 'Y' are P, R, U.
• For each case (UDAP..., UDAR..., UDAU...), we find permutations of the remaining 3 letters.
• Number of permutations for each case = $ 3! = 6 $.
• Total words = $ 3 \times 6 = 18 $.

Permutations Starting 'UDAY' then Before 'P'

• The word starts with 'UDAY'. Remaining letters: {P, R, U}.
• The fifth letter is 'P'. There are no letters smaller than 'P' in the remaining set.
• Total words = $ 0 $.

Permutations Starting 'UDAYP' then Before 'U'

• The word starts with 'UDAYP'. Remaining letters: {R, U}.
• The sixth letter is 'U'. The letter smaller than 'U' is 'R'.
• We fix 'UDAYPR'. Remaining letter: {U} (1 distinct letter).
• Number of permutations = $ 1! = 1 $.
• Total words = $ 1 \times 1 = 1 $.

Permutations Starting 'UDAYPU' then Before 'R'

• The word starts with 'UDAYPU'. Remaining letter: {R}.
• The seventh letter is 'R'. There are no letters smaller than 'R' in the remaining set.
• Total words = $ 0 $.

Final Rank Calculation for UDAYPUR

The rank is the sum of all the words counted before "UDAYPUR", plus 1 for the word itself.

• Rank = $ 1440 + 120 + 0 + 18 + 0 + 1 + 0 + 1 $
• Rank = $ 1580 $.