[f(1+x)^(n)=(a+c,n+c_(2)x^(2)+cdots)(nx^(n)," then "],[sum_(r=0)^(n)sum_(k=0)^(n)(c_(n)+xi)+" equal to "],[Q(n+1)z^(n+1)]

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+….+C_(n)x^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)(r+s)C_(r)C_(s) is equal to :

If a_(n)=sum_(r=0)^(n)(1)/(""^(n)C_(r)) then sum_(r=0)^(n)(r)/(""^(n)C_(r)) equal to

If n in N, then sum_(r=0)^(n) (-1)^(n) (""^(n)C_(r))/(""^(r+2)C_(r)) is equal to .

If n in N, then sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(""^(r+2)C_(r)) is equal to .

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

If a_(n)=sum_(r=0)^(n)(1)/(*^(n)C_(r)), the value of sum_(r=0)^(n)(n-2r)/(n^(n)C_(r))

Statement-1 : sum_(r=0)^(n) r^(2) ""^(n)C_(r) x^(r) = n (n-1) x^(2) (1 + x)^(n-2) + nx (1 +x)^(n-1) Statement-2: sum_(r=0)^(n) r^(2) ""^(n)C_(r) = n (n-1)2^(n-2)+ n2^(n-1) .