Intuitive explanation of "reverse" binomial coefficient formula ${n\choose k}=\sum_{r=0}^s {n-s \choose k-r}{s\choose r}$
32780
Combinatorial Proof Question: Vandermonde convolution identity
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
140877
Evaluate the sum $\sum_{k=1}^n {n \choose k-1} {n \choose k}$
94784
$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
116461
Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
96945
General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$.
143365
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
73050
Explanation of the binomial theorem and the associated Big O notation
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
87479
Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
110441
Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
19914
Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)
59638
How to prove this result on binomial coefficients?
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?