Prove that n choose k is even for 0<k<n if and only if n is a power of 2
99634
$\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.
130093
Prove by Induction : $n < 2^n$
10902
Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
121130
Graph with even vertices and ${n-1}\choose 2$ $+ 1$ edges has a perfect matching
134000
Proof by Strong Induction: $n = 2^a b,\, b\,$ odd, every natural a product of an odd and a power of 2
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
11967
number of subsets of even and odd
6165
Proof by Induction $n^2+n$ is even
90519
Combinatorial proof of $\sum\limits_{k=0}^n {n \choose k}3^k=4^n$
43764
Prove $P_m X P_n$ is Hamiltonian if and only if at least one of $m,n$ is even
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
42744
Prove a number is even using the Pigeonhole Principle
43058
If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
72054
Show that $\deg f$ is even when $n$ is odd.
113002
Any power of $(X_1, X_2,...,X_i)$ is primary in $k[X_1, X_2,...,X_n].$
34929
Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.