If `(1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r` ,then `a_r ` is a.`(.^nC_2)^2` b. `.^n C_r .^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 ^n C_r b. ^n C_r3^r c. ^2n C_r d. ^n C_r2^r

.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)=

""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

For ""^(n) C_(r) + 2 ""^(n) C_(r-1) + ""^(n) C_(r-2) =

""^(n)C_(r+1)+^(n)C_(r-1)+2.""^(n)C_(r)=

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)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.

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)) .

sum_(r=0)^n(-2)^r*(nC_r)/((r+2)C_r) is equal to