17-10-09 Sum of combinatorial Elements.

Find the sum of the given expression here:

\sum_{0\leq i\neq j\leq n}{\left( ^nC_i\times ^nC_j\right)}

2 Answers

106

Asish Mahapatra ·2009-11-09 03:25:15

(ⁿC₀ + ⁿC₁ + ... + ⁿC_n)² = C₀²+C₁²+...+C_n² + S where S excludes the case i=j

=> S = 2²ⁿ - Î£C_i²

(1+x)ⁿ = C₀ + C₁x + .. . + C_nxⁿ
(x+1)ⁿ = C₀xⁿ + C₁x^n-1 + .. . + C_n

Multiplying these two,and equating the coeff of xⁿ,

Î£C_i² = ²ⁿC_n

So, S = 2²ⁿ - ²ⁿC_n

Good work asish :)
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