Understanding a combinatorial relation.

Question

I would like some insight as to why the following expression is true.

$$\sum_{i=0}^n {{n}\choose{i}} 2^{n-i} = 3^n $$

I arrived at this relation in solving a subset problem, and I understand the left hand side in terms of a combinatorial approach to said problem. However, I am having trouble interpreting the right hand side as an equivalent expression, both logically (from a combinatorics perspective) and mathematically.

Answer 1

Note that $$3^n=(1+2)^n=\sum_{i=0}^{n}\binom{n}{i}\cdot 1^{i}\cdot 2^{n-i}=\sum_{i=0}^{n}\binom{n}{i}\cdot 2^{n-i}.$$

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Answer 2

The number of ways that we can color $n$ objects either green, blue, or purple, is equal to the number of ways that we can first select $k$ objects to be green (for arbitrary $k$, $0\leq k \leq n$), then color the rest blue or purple.