Simplify \sum_{i=0}^{n}\binom{n}{i}2^{n-i} to 3^n

[noblerare]

Homework Statement

This is actually part of a larger problem that asks us to prove that the number of ways of counting something is equal to $$3^n$$. I have worked it out and the equation I get is:

$$\binom{n}{0}2^n + \binom{n}{1}2^{n-1}+\ldots+\binom{n}{n}2^{n-n}$$

I am wondering how I should simplify this to make it equal to $$3^n$$

2. The attempt at a solution

I rewrote the above equation into:

$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}2^{n-i}$$

But then I didn't know how to proceed from here since both the combinatorial choosing term and the powered terms are changing. I also tried factoring out $$2^n$$ but that didn't do anything.

Can anyone help me?
Thanks.

 

[tiny-tim]

Hi noblerare!

Hint: what's (2 + x)ⁿ ?

 

[noblerare]

Ohhhh, wow. Okay thanks, tiny-tim! Problem solved.