<p>Let <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> be the increasing sequence of positive integers whose binary representation has exactly <img alt="$8$" height="14" src="https://latex.artofproblemsolving.com/8/4/5/8455f3b5cb3b4880b8c9d782a5c1f0334db819eb.png" width="11" /> ones. Let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> be the 1000th number in <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />. Find the remainder when <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> is divided by <img alt="$1000$" height="13" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="35" />.</p>

Question No: 381

Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$ . Find the remainder when $N$ is divided by $1000$ .

Editorial

Tags: Combinatorics

Question Details