Proof by induction: Sum of binomials

Question

Prove by induction that $$\sum_{i=0}^n\binom ni=2^n\text{ where }\binom ni=\frac{n!}{i!(n-i)!}$$

Please explain me step by step. Thank you.

Answer 1

It is much more easier to prove this without Indcution.

Consider$$(1+x)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}+\binom{n}{2}x^{n-2}+...$$ $$ = \sum _{i=0}^n \binom{n}{i} x^n$$

setting x =1 ,

$$(1+1)^n = \sum _{i=0}^n \binom{n}{i}1^n $$ $$\implies \space \space \space \space\space\space2^n = \sum _{i=0}^n \binom{n}{i} $$