gobathegreat wrote: $a)$ Prove that for all positive integers $n \geq 3$ holds: $$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent $b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers $a)$ use $\binom{n}{0}+\binom{n}{1}+...+\binom{n}{n}=2^n$ $b)$ use $a)$ $\binom{n}{1}+\binom{n}{2}+...+\binom{n}{\frac{n-1}{2}}=\frac{2^n-2}{2}$ When $n \geq 3$ $\frac{2^n-2}{2}$ is odd. Because the sum of all number in the set is odd,the set has odd number of odd numbers.