`sum_(0 leqiltj) sum_( leqn) (C_i+C_j)^2= `

Find the sum (i)sum_(0lei < jlen)sum . ^ nC_i .^ nC_j (ii)sum_(0lei le jlen)sum . ^ nC_i .^ nC_j (iii)sum_(i!=j).^ nC_i .^ nC_j

If C_(0), C_(1), C_(2), …, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(0 ler )^(n)sum_(lt s len)^(n)C_(r)C_(s) =.

If (1 + x)^(n) = sum_(r=0)^(n) C_(r),x^(r) , then prove that (sumsum)_(0leiltjlen) ((i)/(C_(i)) + (j)/(C_(j))) = (n^(2))/(2) sum_(r=0)^(n) (1)/(C_(r)) .

Statement 1: sum sum_(0le ilt j le n)(i/ (^n c_i)+j/(^nc_j)) is equal to(n^2)/2a , where a ,sum_(r="0)^(n) 1/(^n"" c_r)="" .="" statement 2:sum_(r=0)^(n) r/(^n" c_r)="sum_(r=0)^(n)(n-r)/(^n" .

Statement 1:sum_(0<=i<=j)sum_(<=n)C(n,i)C(n,j)=2^(2n)-C(2n,n) Statement 2:(sum_(r=0)^(n))C(n,r))^(2)=sum_(r=0)^(n)(C(n,r))^(2)+2sum_(0<=i<=j)sum_(<=n)C(n,i)C(n,j)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +…+ C_(n) x^(n) , find the values of the following . sum_(i=0)^(n) sum_(j=0)^(n) (i+j) C_(i) C_(j)

Evaluate sum_(0 le i lne j le 10) ""^(21)C_(i) * ""^(21)C_(j) .

Let (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n)[C_(r)=nC_(r)) Statement 1:S=C_(0)+(C_(0)+C_(1))+(C_(0)+C_(1)+C_(2))+...+(C_(0)+C_(1)+...+C_(n))=(n+2)2^(n-1) Statement 2:sum_(i=0)^(n)sum_(j=0)^(n)(C_(i)+C_(j))=(n+1)2^(n)