The letters of the word ``UDAYPUR'' are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word ``UDAYPUR'' is
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In dictionary ranking problems, always add 1 at the end to include the given word itself.
To find the rank of the word "UDAYPUR" when all its letters are arranged in dictionary order, follow these steps:
Identify the letters in "UDAYPUR": U, D, A, Y, P, U, and R.
Determine the alphabetical order: A, D, P, R, U, U, Y.
Calculate the rank of "UDAYPUR" by considering the number of words that can be formed with letters that are lexicographically smaller than the given word.
Let's compute this step-by-step:
Starting with the first letter 'U', the smaller letters before it are A, D, P, and R.
Fix each of these letters at the initial position and count all permutations of the remaining letters U, U, Y:
\(\text{For A: } 6! / 2! = 360\)
\(\text{For D: } 6! / 2! = 360\)
\(\text{For P: } 6! / 2! = 360\)
\(\text{For R: } 6! / 2! = 360\)
Total for letters smaller than 'U' at the first position: \(360 \times 4 = 1440\).
Now with U at the first position, the next letter 'D' can be fixed:
Smaller letters before the next 'Y' are A, P, and R.
Count permutations of the letters A, P, R, with remaining letters U, U, Y:
\(\text{For A: } 5! / 2! = 60\)
\(\text{For P: } 5! / 2! = 60\)
\(\text{For R: } 5! / 2! = 60\)
Total with smaller combinations: \(60 \times 3 = 180\).
Add these counts: \(1440 + 180 = 1620\).
Now consider the words starting with 'U', 'D', 'A' then the next distinguished order 'Y':
Letters for this step: 'P', 'U', 'R', 'U'
'P' is fixed, remaining: U, R, U
\(3! / 2! = 3\)
Add to total: \(1620 + 3 = 1623\).
Add 'R':
Total for this phase: \(1!\) (as only one arrangement is left after 'U' and 'R').
Add final count: \(1\).
Therefore, the rank of the word "UDAYPUR" is 1580.
