If `(1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a=` `(^n C_2)^2` b. `^n C_rdot^n C_(r+1)` c. `^2n C_r` d. `^2n C_(r+1)`

If (1-x^2)^n=sum_(r=0)^n a_r x^r(1-x)^(2n-r),t h e na_r is equal to a.) ^n C_r b.) ^nC_r3^r c.) ^2n C_r d.) ^nC_r2^r

Find the sum sum_(r=0)^n^(n+r)C_r .

Statement1: if n in Na n dn is not a multiple of 3 and (1+x+x^2)^n=sum_(r=0)^(2n)a_r x^r , then the value of sum_(r=0)^n(-1)^r a r^n C_r is zero Statement 2: The coefficient of x^n in the expansion of (1-x^3)^n is zero, if n=3k+1orn=3k+2.

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

If (4x^(2) + 1)^(n) = sum_(r=0)^(n)a_(r)(1+x^(2))^(n-r)x^(2r) , then the value of

The value of determinant |[ ^n C_(r-1), ^n C_r, (r+1)^(n+2)C_(r+1)],[ ^n C_r, ^n C_(r+1),(r+2)^(n+2)C_(r+2)],[ ^n C_(r+1), ^n C_(r+2), (r+3)^(n+2)C_(r+3)]| is n^2+n-2 b. 0 c. ^n+3C_(r+3) d. ^n C_(r-1)+^n C_r+^n C_(r+1)

If (18 x^2+12 x+4)^n=a_0+a_(1x)+a2x2++a_(2n)x^(2n), prove that a_r=2^n3^r(^(2n)C_r+^n C_1^(2n-2)C_r+^n C_2^(2n-4)C_r+) .

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .