Find r > 0 if `.^(n)C_R = ^(n)P_r ,` n being a positive integer.

Consider (1+x+x^(2)) ^(n) = sum _(r=0)^(2n) a_(r) x^(r) , "where " a_(0),a_(1), a_(2),…a_(2n) are real numbers and n is a positive integer. The value of a_(2) is

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. The value of sum_(r=0)^(n-1) a_(r) is

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is odd , the value of sum_(r-1)^(2) a_(2r -1) is

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(2n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is even, the value of sum_(r=0)^(n/2-1) a_(2r) is

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

If "^(n-1)C_r : ^nC_r : ^(n+1)C_r= 6:9: 13 , find n and r.

If .^(n)C_(r )=84, .^(n)C_(r-1)=36 " and" .^(n)C_(r+1)=126 , then find the value of n.

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .

if .^(2n)C_(2):^(n)C_(2)=9:2 and .^(n)C_(r)=10 , then r is equal to