Για n ∈ N ∗ = N − { 0 } {\displaystyle n\in \mathbb {N} ^{*}=\mathbb {N} -{\begin{Bmatrix}0\end{Bmatrix}}} ορίζεται:
n ! = ∏ i = 1 n i {\displaystyle n!=\prod _{i=1}^{n}i} .
Για n = 0 ορίζεται:
0 ! = 1 {\displaystyle 0!=1} .
Για n ∈ N {\displaystyle n\in \mathbb {N} } είναι:
( α + β ) n = ∑ i = 0 n n ! i ! ( n − i ) ! α n − i β i {\displaystyle {\begin{pmatrix}\alpha +\beta \end{pmatrix}}^{n}=\sum _{i=0}^{n}{\frac {n!}{i!{\begin{pmatrix}n-i\end{pmatrix}}!}}\alpha ^{n-i}\beta ^{i}}
Για n ∈ N ∗ = N − { 0 } {\displaystyle n\in \mathbb {N} ^{*}=\mathbb {N} -{\begin{Bmatrix}0\end{Bmatrix}}} και ∀ i ∈ { 1 , 2 , 3... n } ⊂ N ∗ {\displaystyle \forall i\in {\begin{Bmatrix}1,2,3...n\end{Bmatrix}}\subset \mathbb {N} ^{*}} ορίζονται οι διωνυμικοί συντελεστές ως εξής:
( n i ) = n ! i ! ( n − i ) ! = ( n n − i ) {\displaystyle {n \choose i}={\frac {n!}{i!{\begin{pmatrix}n-i\end{pmatrix}}!}}={n \choose n-i}}
Ακόμη ορίζεται:
( n 0 ) = 1 {\displaystyle {n \choose 0}=1} και ( n 1 ) = n {\displaystyle {\binom {n}{1}}=n}
( n i ) + ( n i + 1 ) = ( n + 1 i + 1 ) {\displaystyle {n \choose i}+{n \choose i+1}={n+1 \choose i+1}}
∑ i = 0 n ( n i ) = 2 n {\displaystyle \sum _{i=0}^{n}{n \choose i}=2^{n}}
∑ i = 0 n ( − 1 ) i ( n i ) = 0 {\displaystyle \sum _{i=0}^{n}{\begin{pmatrix}-1\end{pmatrix}}^{i}{n \choose i}=0}
∑ i = 0 m ( n + i n ) = ( n + m + 1 n + 1 ) , m ∈ N {\displaystyle \sum _{i=0}^{m}{n+i \choose n}={n+m+1 \choose n+1},\;m\in \mathbb {N} }
∑ i = 0 k ( n 2 i ) = 2 n − 1 , k = [ n 2 ] {\displaystyle \sum _{i=0}^{k}{n \choose 2i}=2^{n-1},\;k={\begin{bmatrix}{\frac {n}{2}}\end{bmatrix}}}
∑ i = 0 k ( n 2 i + 1 ) = 2 n − 1 , k = [ n − 1 2 ] {\displaystyle \sum _{i=0}^{k}{n \choose 2i+1}=2^{n-1},\;k={\begin{bmatrix}{\frac {n-1}{2}}\end{bmatrix}}}
∑ i = 0 n ( n i ) 2 = ( 2 n n ) {\displaystyle \sum _{i=0}^{n}{n \choose i}^{2}={2n \choose n}}
∑ i = 0 p ( m i ) ( n p − i ) = ( m + n p ) , m ∈ N , p ≤ m i n { m , n } ∧ p ∈ N {\displaystyle \sum _{i=0}^{p}{m \choose i}{n \choose p-i}={m+n \choose p},\;m\in \mathbb {N} ,\;p\leq min{\begin{Bmatrix}m,n\end{Bmatrix}}\wedge p\in \mathbb {N} }
∑ i = 1 n i ( n i ) = n 2 n − 1 {\displaystyle \sum _{i=1}^{n}i{n \choose i}=n2^{n-1}}
∑ i = 1 n ( − 1 ) i + 1 i ( n i ) = 0 {\displaystyle \sum _{i=1}^{n}{\begin{pmatrix}-1\end{pmatrix}}^{i+1}i{n \choose i}=0}
( p ∑ i = 1 α i ) n = ∑ i = 1 p n ! p ∏ i = 1 n i ! ∏ i = 1 p α i n i , p ∈ N ∗ , n i ∈ N ∗ ∧ ∑ i = 1 p n i = n {\displaystyle {\begin{pmatrix}{\begin{matrix}p\\\sum \\i=1\end{matrix}}\alpha _{i}\end{pmatrix}}^{n}=\sum _{i=1}^{p}{\frac {n!}{{\begin{matrix}p\\\prod \\i=1\end{matrix}}n_{i}!}}\prod _{i=1}^{p}\alpha _{i}^{n_{i}},\;p\in \mathbb {N} ^{*},\;n_{i}\in \mathbb {N} ^{*}\wedge \sum _{i=1}^{p}n_{i}=n}
