JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Permutations and Combinations

The seven letters of UDAYPUR in dictionary order are $A,D,P,R,U,U,Y$ (with $U$ repeated), giving $\frac{7!}{2!}$ arrangements. Counting words that precede UDAYPUR: a first letter from $\{A,D,P,R\}$ gives $4\times \frac{6!}{2!}=4\times 360=1440$. Fixing $U$ first, a second letter $A$ (before $D$) gives $5!=120$. Fixing $UDA$, the fourth letter being any of $\{P,R,U\}$ before $Y$ gives $3\times 3!=18$. Fixing $UDAYP$, a sixth letter $R$ before $U$ gives $1!=1$. Adding $1$ for the word itself, the rank is $1440+120+18+1+1=1580$.