Simplified formula for calculating sum of $\binom{n_1}{r_1} \cdot \binom{n_2}{r_2} + \cdots$.
17580
Combinatorics That Looks Similar to Vandermonde's Identity
68848
How to simplify $\sum_{k=1}^nk\cdot k!$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
53116
How find this sum $f(n)=\sum_{i=1}^{n}\dfrac{\binom{n}{i}}{i}$
114173
Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$
110441
Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
64803
Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
14306
What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$
124135
A general formula for $\sum (k-1)(k-2)(k-3)$?
96945
General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$.
87589
Evaluate $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...+\binom{n}{2k}$
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
60860
Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
83229
Nested summations and their relation to binomial coefficients
65648
Different ways to come up with $1+2+3+\cdots +n=\frac{n(n+1)}{2}$
94784
$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument
94060
Sum of square binomial coefficients