let-give-n-p-from-N-2-and-1-p-n-prove-that-k-0-p-C-n-k-C-n-k-p-k-2-p-C-n-p-

Tinku Tara June 4, 2023 Arithmetic 0 Comments

Question Number 29163 by abdo imad last updated on 04/Feb/18

let give (n,p) from N^2 and 1≤p≤n prove that Σ_(k=0) ^p C_n ^k C_(n−k) ^(p−k) ==2^p C_n ^p .

$${let}\:{give}\:\left({n},{p}\right)\:{from}\:{N}^{\mathrm{2}} \:{and}\:\mathrm{1}\leqslant{p}\leqslant{n}\:{prove}\:{that}\: \\ $$$$\sum_{{k}=\mathrm{0}} ^{{p}} \:{C}_{{n}} ^{{k}} \:{C}_{{n}−{k}} ^{{p}−{k}} ==\mathrm{2}^{{p}} \:\:{C}_{{n}} ^{{p}} . \\ $$$$ \\ $$

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