Let $N,E,X,K>0$ be integers such that $N+E<X$ and $K\le\min(N,E)$. The author proves the inequality $${X\choose N}-\sum_{j=0}^K{X-E\choose N-j}{E\choose j}\le{X\choose N}{NE\over (K+1)X},$$ where the motivation is that the left-hand side counts those $N$-element subsets of an $X$-element set $S$ which intersect a fixed $E$-element subset $T\subset S$ in at least $K+1$ elements. The inequality is compared with another one of the same sort that was obtained by Brüdern and Perelli.
Counting sets with exceptions / G. Molteni. - In: MATHEMATICAL INEQUALITIES & APPLICATIONS. - ISSN 1331-4343. - 7:2(2004), pp. 161-164.
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