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If C(0),C(1),C(2)…….,C(n) are the combin...

If `C_(0),C_(1),C_(2)…….,C_(n)` are the combinatorial coefficient in the expansion of `(1+x)^n, n, ne N`, then prove that following
`C_(1)+2C_(2)+3C_(3)+..+n.C_(n)=n.2^(n-1)`
`C_(0)+2C_(1)+3C_(2)+......+(n+1)C_(n)=(n+2)C_(n)=(n+2)2^(n-1)`
` C_(0),+3C_(1)+5C_(2)+.....+(2n+1)C_n =(n+1)2^n`
`(C_0+C_1)(C_1+C_2)(C_2+C_3)......(C_(n-1)+C_n)=(C_0.C_1.C_2....C_(n-1)(n+1)^n)/(n!)`
`1.C_0^2+3.C_1^2+....+ (2n+1)C_n^2=((n+1)(2n)!)/(n! n!)`

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