Prove that the number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of $n+k$ into exactly $k$ parts.
92941
# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
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.
139078
Infinite partition of $\mathbb N$ by infinite subsets
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
84010
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
117862
Number of partitions of an $n$-element set into $k$ classes
128716
Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side
43979
The total number of subsets is $2^n$ for $n$ elements
53937
Any partition of $\{1,2,\ldots,100\}$ into seven subsets yields a subset with numbers $a,b,c,d$ such that $a+b=c+d$.
79984
Partition of ${1, 2, ... , n}$ into subsets with equal sums.
59535
Prove that there exists such a set of $n$ positive integers
103538
Partitions of $n$ into distinct odd and even parts proof
74730
Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
64527
Show that an operation of elements in a finite group contains each element exactly once
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
48838
Proving the number of $n$ length binary strings with no consecutive $1's$ $b_n$ is equal to $b_{n-1} + b_{n-2}$