Question

$\: \: \: \: \: \: \: \: \: \: \: n\\ evaluate \: \: \: \: \: Σ \: (nCi)\\ \: \: \: \: \: \: \: \: \: \: \: \: i = 0$
Evaluate the summation​

Answer 1

Assuming you mean

$\displaystyle \sum_{i=0}^n {}_nC_{i}$

where

${}_n C_i = \dbinom ni = \dfrac{n!}{i! (n-i)!}$

we have by the binomial theorem

$\displaystyle (1 + 1)^n = \sum_{i=0}^n {}_nC_{i} \cdot 1^i \cdot 1^{n-i}$

so that the given sum has a value of $\boxed{2^n}$.