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If(1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following (sumsum)_(0leile jlen)(i +j)(C_(i)pmC_(j) )^(2) |
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Answer» SOLUTION :` underset(0leiltjlen)(sumsum) (i + j) C_(i) C_(j)` Let ` P = underset(0leiltjlen)(sumsum)(i + j) C_(i) C_(j)`...(i) Replacing I by n - I and j by n- j in Eq.(i), then we GET ` P=underset(0leiltjlen)(sumsum) (n-i+ n-j)C_(n-1)C_(n-1) ` `[because ` SUM of BINOMIAL expansion does not change if we replace r byn - r ] ` P=underset(0leiltjlen)(sumsum) (2N -i - j) C_(i) C_(j)"" [because ""^(n)C_(r) = ""^(n)C_(n-1)]` ...(ii) On addingEqs.(i) and (ii) , we get ` 2P=2n underset(0leiltjlen)(sumsum)C_(i) C_(j)` or `P= n underset(0leiltjlen)(sumsum) C_(i) C_(j) = n/2(2^(2n) -""^(2n)C_(n))""`[from corollary I] |
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