Show that the number of possible compositions for n ∈ $\mathbb{N}$ is $2^{n−1}$
73224
number of ordered partitions of integer
4970
For $n \geq 2$, show that $n \nmid 2^{n}-1$
26514
Prove that if $n$ is a composite, then $2^n-1$ is composite.
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.
39724
Combinatorial Identities
82571
Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
129304
Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
37012
The number of summands $\phi(n)$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
46971
Number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$
20964
Proof that $n^2 < 2^n$
15000
Show that number of elements in $(Z/nZ)^*$ is $\phi (n)$
11867
Prove by induction that $n < 2^n$ for all $n \ge 1$
35553
If $n$ is composite, so is $2^n-1$
150156
Proving that $S_n$ has order $n!$
88628
Proving the total number of subsets of S is equal to $2^n$