Prove that ${{n+m-1}\choose{m}}$ is the number of ways to select m items from a set of n items with repetitions allowed.
116647
The number of different ways to choose $k$ out of $n$ unique items
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
34452
In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive?
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
55840
Number of functions from n-element set to {1, 2, …, m}
43979
The total number of subsets is $2^n$ for $n$ elements
120642
Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
41354
Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
67927
Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
76635
Proof through combinatorial argument
75461
If $n>m$, then the number of $m$-cycles in $S_n$ is given by $\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}$.
925
Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
136037
${n\choose 2} - {k\choose 2} = {n-k\choose 2} +k(n-k)$ Marked box or unmarked box
44484
Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$
89611
Combinations of selecting $n$ objects with $k$ different types
120074
Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$