Combinatorial Proof Of ${n \choose k}={n-1\choose {k-1}}+{n-1\choose k}$
424
Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$
31196
Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
90904
Combinatorial proof of an identity with squared binomial coefficients
81729
Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
117775
Combinatorial proof of a Stirling number identity.
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
4610
Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
156566
Combinatorial proof of a binomial identity: $\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$
121279
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
29828
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
136037
${n\choose 2} - {k\choose 2} = {n-k\choose 2} +k(n-k)$ Marked box or unmarked box
10211
Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$