Find `underset(r=0) overset(10)sumr^ (10)C_(r).3^(r).(-2)^(10-r)`

The value of underset(r=0)overset(40)sumr.^(40)C_(r).^(30)C_(r) is

Statement-1: In the expansion of (sqrt(5)+3^(1//5))^(10) , sum of integral terms is 3134. Statement-2: (x+y)^(n)=underset(r=0)overset(n)(sum).^(n)C_(r)*x^(n-r)y^(r) .

If (1-x^(3))^(n)=underset(r=0)overset(n)(sum)a_(r)x^(r)(1-x)^(3n-2r) , then the value of a_(r) , where n in N is

underset(r=0)overset(n)(sum)(-1)^(r).^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))+ . . .m" terms"]=

let underset(r=0)overset(2010)(sum)a_(r)x^(r)=(1+x+x^(2)+x^(3)+x^(4)+x^(5))^(402) and underset(r=0)overset(2010)(sum)a_(r)=a, then the value of ((underset(r=0)overset(2010)(sum)r.a_(r))/(underset(r=0)overset(2010)(sum)a_(r))) is equal to____.

Statement-1: If underset(r=1)overset(n)(sum)r^(3)((.^(n)C_(r))/(.^(n)C_(r-1)))^(2)=196 , then the sum of the coeficients of powerr of xin the expansion of the polynomial (x-3x^(2)+x^(3))^(n) is -1. Statement-2: (.^(n)C_(r))/(.^(n)C_(r-1))=(n-r+1)/(r) AA n in N and r in W .

The sum of the series underset(r = 0)overset(10)Sigma .^(20)C_(r) " is " 2^(19) + (.^(20)C_(10))/(2)

If n is a positive integer, prove that underset(r=1)overset(n)sumr^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

underset(r=1)overset(n)(sum)r(.^(n)C_(r)-.^(n)C_(r-1)) is equal to

If S_(n) = underset (r=0) overset( n) sum (1) /(""^(n) C_(r)) and T_(n) = underset(r=0) overset(n) sum (r )/(""^(n) C_(r)) then (t_(n))/(s_(n)) = ?