for $N \in \mathbb N$ and $k \in \mathbb N$ prove that there exist unique $0\le x_1 \lt x_2\lt \dots \lt x_k$ such that $$N = \sum_{i=1}^k \binom{x_i}{i}$$
edit: i've tried looking for combintorial proof but i couldnt find the right interpretation since $x$-es are not given.
then i tried looking at right pascal triangle and tried a few numbers, I found out that i always take the biggest possible $x_k$ for $k$, then the biggest possible $x_{k-1}$ for $k-1$ sort of how you find binary expression for natural numbers.