Exercise 6, § 5

Tag 03MI, permanent. Cite it as [Bourbaki, Tag 03MI].

Theory of Sets, III, 236

Prove the formula

\binom{n+h}{h} = 1+ \binom{h}{1}\binom{n+h-1}{h} - \binom{h}{2}\binom{n+h-2}{h} +\cdots +(-1)^h\binom{h}{h}\binom{n}{h}.

(If F denotes the set of mappings $u$ of $[1,h]$ into $[0,n]$ such that

\sum_{x=1}^{h}u(x)\leq n,

consider for each subset H of $[1,h]$ the set of all $u\in F$ such that $u(x)\geq 1$ for each $x\in H$ , and use Exercise 5.)

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