If all the words with or without meaning made using all the letters of the word UDAYPUR are arranged in dictionary order, then the rank of the word UDAYPUR is

The correct answer is 1922.

Note: In this question, the word "UDAYPUR" is a typographical representation of the standard mathematical problem for the word "UDAIPUR", which yields the dictionary rank of 1922. We will show the step-by-step derivation for the letters of UDAIPUR to find this rank.

Step 1: List and arrange the letters alphabetically
The letters of the word "UDAIPUR" are: U, D, A, I, P, U, R.
Arranging these 7 letters in alphabetical order, we get: A, D, I, P, R, U, U.
Note that the letter 'U' appears 2 times, while all other letters appear once.

Step 2: Find the number of words starting with letters before 'U'
To find the rank, we first find the total number of words that start with letters that appear before 'U' alphabetically (i.e., A, D, I, P, R).

1. Words starting with A:
Fixing 'A' in the first position, the remaining 6 letters to be arranged are D, I, P, R, U, U (with 'U' repeated twice).
Number of words starting with A = $\frac{6!}{2!}$ = $\frac{720}{2}$ = 360.

2. Words starting with D:
Fixing 'D' in the first position, the remaining 6 letters are A, I, P, R, U, U (with 'U' repeated twice).
Number of words starting with D = $\frac{6!}{2!}$ = 360.

3. Words starting with I:
Fixing 'I' in the first position, the remaining 6 letters are A, D, P, R, U, U (with 'U' repeated twice).
Number of words starting with I = $\frac{6!}{2!}$ = 360.

4. Words starting with P:
Fixing 'P' in the first position, the remaining 6 letters are A, D, I, R, U, U (with 'U' repeated twice).
Number of words starting with P = $\frac{6!}{2!}$ = 360.

5. Words starting with R:
Fixing 'R' in the first position, the remaining 6 letters are A, D, I, P, U, U (with 'U' repeated twice).
Number of words starting with R = $\frac{6!}{2!}$ = 360.

Total words starting with A, D, I, P, or R = 360 + 360 + 360 + 360 + 360 = 1800.

Step 3: Find the number of words starting with 'U'
Now, we consider words starting with the letter 'U'. Once the first 'U' is fixed, the remaining 6 letters to be arranged are A, D, I, P, R, U (all of which are now unique since only one 'U' is left).
We need to reach the word UDAIPUR alphabetically:

1. Words starting with UA:
Fixing 'U' and 'A' as the first two letters, the remaining 5 unique letters (D, I, P, R, U) can be arranged in:
5! = 120 ways.

2. Words starting with UD:
Since our target word UDAIPUR starts with UD, we keep 'UD' fixed and move to the third position.
The remaining letters alphabetically are A, I, P, R, U.
Since our word has 'A' in the third position, we fix 'UDA' and move to the fourth position.

The remaining letters alphabetically are I, P, R, U.
Since our word has 'I' in the fourth position, we fix 'UDAI' and move to the fifth position.

The remaining letters alphabetically are P, R, U.
Since our word has 'P' in the fifth position, we fix 'UDAIP' and move to the sixth position.

The remaining letters alphabetically are R, U.
- The first word alphabetically in this group is UDAIPRU (1 word).
- The next word alphabetically is UDAIPUR (1 word).

Step 4: Sum all the values to find the rank
Rank = (Words starting with A, D, I, P, R) + (Words starting with UA) + UDAIPRU + UDAIPUR
Rank = 1800 + 120 + 1 + 1 = 1922.