Simplifying an expression with binomial coefficients and powers of $2$

Question

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

Anyway to simplify this such that it can become of 'closed' form (i.e. a concrete number of terms)?

Answer 1

Recall the binomial theorem:

$$(a+b)^n = \sum_{k=0}^n \binom{n}{k} \cdot a^k \cdot b^{n-k}$$

Notice that, if you have $a=1, b=2$, then the sum you're trying to get is on the right side. What appears on the left?

Answer 2

I think it's $$(2+1)^n$$ by the binom of Newton.