The smallest $n$ for which the sum of binomial coefficients exceeds $31$

Question

I have a problem with the binomial theorem.

What is the result of solving this inequality:

$$ \binom{n}{1} + \binom{n}{2} + \binom{n}{3} + \cdots +\binom{n}{n} > 31 $$

Answer 1

HINT:

For any positive integer $n,$

$$\sum_{r=1}^n\binom nr=(1+1)^n-1$$

Answer 2

Since we have $$\sum_{k=0}^{n}\binom{n}{k}=\sum_{k=0}^{n}\binom{n}{k}\cdot 1^{n-k}\cdot 1^k=(1+1)^n=2^n,$$ we have $$\sum_{k=1}^{n}\binom{n}{k}\gt 31\iff\sum_{k=\color{red}{0}}^{n}\binom{n}{k}\gt 32\iff 2^n\gt 2^5\iff n\gt 5.$$