For all positive integers n,m,k where $n\ge m\ge k$ , $\binom {n}{m}\binom {m}{k}=\binom {n}{k}\binom {n-k}{n-m}$
31196
Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
67650
Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$
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
64803
Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
87813
How to show that for all k, $k! \ge (k/2)^{k/2}$
115258
Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
127855
Show that the product of any $m$ consecutive positive integers is divisible by $m!$
129304
Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
25004
Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
107335
Prove equality between binomial coefficients.
102174
Proving that all terms in sequence are positive integers (Recursive)
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
147079
$\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$