JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Permutations and Combinations

There are $5^5=3125$ such strings, and the serial number of $35337$ in descending order is one more than the number of strings that exceed it. Comparing digit by digit, with the ranking $7>5>3>2>1$: the first digit $3$ has $5,7$ above it, contributing $2\cdot 5^4=1250$; after a leading $3$, the second digit $5$ has $7$ above it, $1\cdot 5^3=125$; after $35$, the third digit $3$ has $5,7$ above, $2\cdot 5^2=50$; after $353$, the fourth digit $3$ has $5,7$ above, $2\cdot 5=10$; after $3533$, the last digit $7$ has nothing above it. So $1250+125+50+10=1435$ strings exceed $35337$, and its serial number is $1436$.