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1) C1+2C2+3C3+--------+nCn=n2 to power n-1

$C_{1} + 2 C_{2} + 3 C_{3} + ............ + n C_{n} = n + 2 \times \frac{n (n - 1)}{2!} + 3 \times \frac{n (n - 1) (n - 2)}{3!} + ............. + n \times 1 = n + 2 \times \frac{n (n - 1))}{2} + 3 \times \frac{n (n - 1))}{6} + ................... + n = n + \frac{n (n - 1)}{1} + \frac{n (n - 1))}{2} + ............... + n = n [1 + \frac{n - 1}{1} + \frac{n (n - 1)}{2} + .......... + 1] = n [1 + \frac{n - 1}{1!} + \frac{n (n - 1)}{2!} + .......... + 1] P u t n - 1 = N = n [1 + \frac{N}{1!} + \frac{N (N + 1)}{2!} + .......... + 1] = n ({}^{N}{C_{0}} + {}^{N}{C_{1}} + {}^{N}{C_{2}} + ............ + {}^{N}{C_{N}}) = n . 2^{N} = n . 2^{n - 1} = R . H . S . H e n c e p r o v e d .$

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