Prove that ${2^n-1\choose k}$ and ${2^n-k\choose k}$ ar always odd.
75723
Odd Binomial Coefficients?
89474
every permutation is either even or odd,but not both
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
18633
Prove every odd integer is the difference of two squares
84010
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
52089
Group of $2n$ elements, $n$ odd, is not simple
75910
Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
44484
Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$
143543
Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
44182
Proving that an integer is even if and only if it is not odd
129238
Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
92057
Prove that for every odd integer n, there is a group...
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
103538
Partitions of $n$ into distinct odd and even parts proof