Show that `C_0 + (C_0 + C_1) + (C_0 +C_1 + C_2)+……. +`
`(C_0 + C_1 + …+C_n)= (n+2).2^(n-1)`

If (1+x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + "….." + C_(n)x^(n) , then C_(0) - (C_(0) + C_(1)) +(C_(0) + C_(1) + C_(2)) - (C_(0) + C_(1) + C_(2) + C_(3))+ "….." (-1)^(n-1) (C_(0) + C_(1) + "……" + C_(n-1)) is (where n is even integer and C_(r) = .^(n)C_(r) )

3. C_0 + 7. C_1 + 11. C_2 +…...+ (4n+3) . C_n=

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

C_0 + 2.C_1 + 4.C_2 + …….+C_n.2^n = 243 , then n =

If C_(r) = ""^(n)C_(r) and (C_(0) + C_(1)) (C_(1) + C_(2)) … (C_(n-1) + C_(n)) = k ((n +1)^(n))/(n!) , then the value of k, is

Prove that C_0 + 2.C_1 + 4.C_2 + 8.C_3 + ……+2^n.C_n = 3^n

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) + 2C_(1) + 3C_(2) + …+ (n+1)C_(n) = (n+2)2^(n-1) .

C_0C_2 + C_1C_3 +C_2C_4+……..+C_(n-2) C_n =

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " 1^(2)*C_(1) + 2^(2) *C_(2) + 3^(2) *C_(3) + …+ n^(2) *C_(n) = n(n+1)* 2^(n-2) .