Koefiziente binomial

Konbinatorian, koefiziente binomiala [n] multzoak duen k tamainako azpimultzo kopurua.

$C_{n}={\binom {n}{k}}={\frac {n!}{k!(n-k)!}},n,k\geq {0}$

Gainera, Newtonen Binomioaren formula erabiliz, koefiziente binomiala $(1+x)^{n}$ polinomioan $x^{k}$ monomioaren koefiziente da.

(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}

Kalkulua

n zenbaki oso ez negatiboa eta k zenbaki oso bat izanik, koefiziente binomiala honela definitutako zenbaki arrunta da:

{n \choose k}={\frac {n\cdot (n-1)\cdots (n-k+1)}{k\cdot (k-1)\cdots 1}}={\frac {n!}{k!(n-k)!}}\quad {\mbox{ ; }}\ n\geq k\geq 0\qquad (1)

${\binom {n}{0}}=1={\binom {n}{n}}$
Baldin eta $k>n$ bada, ${\binom {n}{k}}=0$
Baldin eta $0\leq k\leq n$ bada, ${\binom {n}{k}}={\binom {n}{n-k}}$
${\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-1}{k-1}}$
Baldin eta $0\leq r\leq k\leq n$ bada, ${\binom {n}{k}}{\binom {k}{r}}={\binom {n}{r}}+{\binom {n-r}{k-r}}$
Baldin eta $1\leq k\leq n$ bada, ${\binom {n}{k}}={\frac {n}{k}}{\binom {n-1}{k-1}}$
Baldin eta $1\leq k\leq n$ bada, ${\binom {n}{k}}={\frac {n-k+1}{k}}{\binom {n}{k-1}}$
${\binom {n}{0}},{\binom {n}{1}},{\binom {n}{2}},...,{\binom {n}{n}},$ seguida gorakorra da bere maximoraino eta gero beherakorra. Baldin n bikoitia bada, maximoa ${\binom {n}{\frac {n}{2}}}$ da; bestela maximoak ${\binom {n}{\frac {n-1}{2}}}$ eta ${\binom {n}{\frac {n+1}{2}}}$ dira.

${\binom {n}{0}}+{\binom {n}{1}}+{\binom {n}{2}}+...+{\binom {n}{n}}=2^{n}$
${\binom {n}{0}}+{\binom {n}{2}}+{\binom {n}{4}}+...={\binom {n}{1}}={\binom {n}{3}}+{\binom {n}{5}}+...=2^{n}-1$
${\binom {n}{0}}-{\binom {n}{1}}+{\binom {n}{2}}-...+(-1)^{n}{\binom {n}{n}}=0,n\geq 1$
${\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-2}{k-1}}+{\binom {n-3}{k-2}}...+{\binom {n-(k+1)}{0}}$
${\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-2}{k-1}}+{\binom {n-3}{k-1}}...+{\binom {k-1}{k-1}}$
Vandermonderen identitatea. Izan bitez $m,n,r\geq 0$ , ${\binom {m+n}{r}}=\sum _{k=0}^{r}{\binom {m}{k}}{\binom {n}{r-k}}={\binom {m}{0}}{\binom {n}{r}}+{\binom {m}{1}}{\binom {n}{r-1}}+{\binom {m}{2}}{\binom {n}{r-2}}+...+{\binom {m}{r}}{\binom {n}{0}}$
${\binom {2n}{n}}=\sum _{k=0}^{n}{\binom {n}{k}}^{2}$

{7 \choose 3}={\frac {7!}{3!(7-3)!}}={\frac {7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{(3\cdot 2\cdot 1)(4\cdot 3\cdot 2\cdot 1)}}={\frac {7\cdot 6\cdot 5}{3\cdot 2\cdot 1}}=35.

Koefiziente binomialak (x + y)ⁿ binomioaren garapeneko koefizienteak dira (hortik datorkio bere izena):

(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}.\qquad (2)