#1433 · Algebra · Linear equations

Sum of products of binomial coefficients

If $c_{0}, c_{1}, c_{2}, \dots, c_{n}$ denote the coefficients in the expansion of $(1 + x)^{n}$ , prove that $c_{0} c_{r} + c_{1} c_{r + 1} + c_{2} c_{r + 2} + \dots + c_{n - r} c_{n} = \frac{( 2 n )!}{( n - r )! ( n + r )!}$ .

~15 minProof · checked by AIBinomial TheoremCombinatorial Identities

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIII. Binomial Theorem: Positive Integral Index, Examples XIII.b, No. 23

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