$\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
50413
Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
110876
Prove that $p\mid \binom{p}{k},\ 0< k< p$
108401
Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
45171
How prove this $(p-1)!\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\right)\equiv 0\pmod{p^2}$
70248
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
22962
How to show $\binom{2p}{p} \equiv 2\pmod p$?
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
51465
New, elegant proofs for $\varphi(p^{k})=p^{k}-p^{k-1}$
158508
Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$
132984
$\sum\limits_{k=0}^{\frac{p-1}2}3^k\binom{p}{k}\equiv 2^p-1\pmod{p^2}$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$