Exercise 3, § 5

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

Theory of Sets, III, 235

Prove the relations

\binom{n}{0}\binom{n}{p} + \binom{n}{1}\binom{n-1}{p-1} + \binom{n}{2}\binom{n-2}{p-2} + \cdots + \binom{n}{p}\binom{n-p}{0} = 2^p\binom{n}{p},

\binom{n}{0}\binom{n}{p} - \binom{n}{1}\binom{n-1}{p-1} + \binom{n}{2}\binom{n-2}{p-2} - \cdots + (-1)^p\binom{n}{p}\binom{n-p}{0} = 0.

(Consider the subsets of $p$ elements of $[1, n]$ which contain a given subset of $k$ elements $(0 \leqslant k \leqslant p)$ , and use Exercise 2 for the second formula.)

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Exercise 7, § 5

Cites

Exercise 2