`C(n,0)+C(n,1)+C(n,2)+...+C(n,n)=`

The value of C(n, 0) - C(n, 1) + C(n, 2) - C(n, 3) +.......+(-1)^(n)C(n, n) =

The mean of the values of 0, 1, 2, …., n, having corresponding weight .^(n)C_(0), .^(n)C_(1), .^(n)C_(2), ....., .^(n)C_(n) respectively is :

If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

If (1+x)^(n) = C_(0)+C_(1).x+C_(2). x^(2)+..+C_(n). x^(n) then C_(0)+2. C_(1)+3. C_(2)+..+(n+1). C_(n) =

The arithmetic mean of ""^(n)C_(0),""^(n)C_(1),""^(n)C_(2), ..., ""^(n)C_(n) , is

The arithmetic mean of ""^(n)C_(0),""^(n)C_(1),""^(n)C_(2), ..., ""^(n)C_(n) , is

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .