MathDB
Proving a combinatorial identity and finding expression

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October 30, 2010
inductioncombinatorics proposedcombinatorics

Problem Statement

If Cnp=n!p!(n−p)!(p≥1)C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1), prove the identity Cnp=Cn−1p−1+Cn−2p−1+⋯+Cpp−1+Cp−1p−1C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1} and then evaluate the sum S=1⋅2⋅3+2⋅3⋅4+⋯+97⋅98⋅99.S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.
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