JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Permutations and Combinations

A five-digit number greater than 50000, built from $0,1,\dots ,7$ with repetition allowed, must start with $5,6$ or $7$.

For the sum of first and last digit to not exceed 8, the (first, last) pairs are:

first $=5$: last $\in \{0,1,2,3\}$ — 4 choices

first $=6$: last $\in \{0,1,2\}$ — 3 choices

first $=7$: last $\in \{0,1\}$ — 2 choices

Total (first, last) pairs $=4+3+2=9$

The three middle digits are unrestricted, each with 8 choices: $8^3=512$

Total count $=9\times 512=4608$

This count includes $50000$ itself (first $=5$, last $=0$, middle $=000$), which is not greater than 50000, so it must be removed:

$4608-1=4607$