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 = (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}`.