Prove that coefficient of `x^(r )` in `(1-x)^(-n) " is " ""^(n+r-1)C_(r )`

Prove that the coefficient of x^r in the expansion of (1-2x)^(-1/2) is (2r!)/[(2^r)(r!)^2]

If C_(r) be the coefficients of x^(r) in (1 + x)^(n) , then the value of sum_(r=0)^(n) (r + 1)^(2) C_(r) , is

Sum of coefficients of x^(2r) , r = 1,2,3…. in (1+x)^n is

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r for which S = .^(20)C_(r.).^(10)C_(0)+.^(20)C_(r-1).^(10)C_(1)+"........".^(20)C_(0).^(10)C_(r) is maximum can not be

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r(0 le r le 30) for which S = .^(20)C_(r).^(10)C_(0) + .^(20)C_(r-1).^(10)C_(1) + ........ + .^(20)C_(0).^(10)C_(r) is minimum can not be

how that the coefficient of (r+1) th in the expansion of (1+x)^(n+1) is equal to the sum of the coefficients of the r th and (r+1) th term in the expansion of (1+x)^n

If the coefficient of r^(th) ,( r +1)^(th) " and " (r +2)^(th) terms in the expansion of (1+x)^n are in A.P then show that n^2 - (4r +1)n + 4r^2 - 2 =0

If in the expansion of (1-x)^(2n-1) a_r denotes the coefficient of x^r then prove that a_(r-1) +a_(2n-r)=0

The coefficient of x^n in (1+x)^(101)(1-x+x^2)^(100) is non zero, then n cannot be of the form a. 3r+1 b. 3r c. 3r+2 d. none of these

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))