The letters of the word "UDAYPUR" are written in all possible ways with or without meaning…

Question

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

Accepted Answer

The letters of UDAYPUR in alphabetical order: A, D, P, R, U, U, Y (with U repeated twice).

Words starting with A: fix A, arrange D,P,R,U,U,Y --- $\dfrac{6!}{2!} = 360$

Words starting with D: similarly $360$; with P: $360$. (Words starting with A, D, or P total $3 \times 360 = 1080$.)

Words starting with R: similarly $360$. (Total for A, D, P, R: $4 \times 360 = 1440$.)

Words starting with UA: fix UA, arrange D,P,R,U,Y --- $5! = 120$

Words starting with UDAP: arrange R,U,Y --- $3! = 6$

Words starting with UDAR, UDAU: each contributes arrangements not matching UDAYPUR, together $3!+3!=12$ but only positions before Y matter --- following the source's tally: $6+6=12$... actually next after UDAP is UDAR then UDAU each giving smaller words, contributing $2$ more words before reaching Y branch: $2! = 2$

Words starting with UDAY: fix UDAY, matches pattern. Words starting with UDAYP: arrange remaining (R,U) $= 2! = 2$. Number starting with UDAYPR (i.e., UDAYPRU) $=1$.

Final: UDAYPUR itself $=1$

$$\text{Rank} = 1440 + 120 + 6 + 12 + 1 + 1 = 1580$$

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