Koefiziente binomial

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

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

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

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

Kalkulua

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

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

Oinarrizko propietateak

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

Koefiziente binomialen identitateak

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

Adibidea

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

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

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

Ikus, gainera

• Konbinazio (konbinatoria)

Kanpo estekak

• Datuak: Q209875
• Multimedia: Binomial coefficients / Q209875