Pascalen hirukia koefiziente binomialak erraz kalkulatzeko erabiltzen da Konbinatorian, koefiziente binomiala [n] multzoak duen k tamainako azpimultzo kopurua.
Gainera, Newtonen Binomioaren formula erabiliz, koefiziente binomiala
polinomioan
monomioaren koefiziente da.
![{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c80c2502413b225d48e79a475156a5f7b677e21)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/073134ae87de7758dfb4c070441b34ef9ac48ec4)
Oinarrizko propietateak
![{\displaystyle {\binom {n}{0}}=1={\binom {n}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac056fde6b2ae35510640bf60507551b19788a3c)
- Baldin eta
bada, ![{\displaystyle {\binom {n}{k}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2bc3e21cb9f37945844d08f67b177b20cabda3)
- Baldin eta
bada, ![{\displaystyle {\binom {n}{k}}={\binom {n}{n-k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1be4c4c9a68f088f7e13fe0b16135695cc638c)
![{\displaystyle {\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-1}{k-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/988ca457c5d2294950caf31c751ffe316768fd0d)
- Baldin eta
bada, ![{\displaystyle {\binom {n}{k}}{\binom {k}{r}}={\binom {n}{r}}+{\binom {n-r}{k-r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3053ff712f56b3cb5221c9be58abab212fc355)
- Baldin eta
bada, ![{\displaystyle {\binom {n}{k}}={\frac {n}{k}}{\binom {n-1}{k-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea308b9a8b378888dff688405b4de68b07eaa9f)
- Baldin eta
bada, ![{\displaystyle {\binom {n}{k}}={\frac {n-k+1}{k}}{\binom {n}{k-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a0f33c97305d66f37e437d7217237e5a7d50c2)
seguida gorakorra da bere maximoraino eta gero beherakorra. Baldin n bikoitia bada, maximoa
da; bestela maximoak
eta
dira.
Koefiziente binomialen identitateak
![{\displaystyle {\binom {n}{0}}+{\binom {n}{1}}+{\binom {n}{2}}+...+{\binom {n}{n}}=2^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7e2ca46bbdb41f41fabd2c4655e1b80aaeec58)
![{\displaystyle {\binom {n}{0}}+{\binom {n}{2}}+{\binom {n}{4}}+...={\binom {n}{1}}={\binom {n}{3}}+{\binom {n}{5}}+...=2^{n}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5f3f732c258d32aa391def9e01ab2d1ca256ae)
![{\displaystyle {\binom {n}{0}}-{\binom {n}{1}}+{\binom {n}{2}}-...+(-1)^{n}{\binom {n}{n}}=0,n\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cea6ae710081eb0a27a506889c31876f69518be)
![{\displaystyle {\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-2}{k-1}}+{\binom {n-3}{k-2}}...+{\binom {n-(k+1)}{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/020895cac7bafb04ffe65fb033eab52eaeb2c170)
![{\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-2}{k-1}}+{\binom {n-3}{k-1}}...+{\binom {k-1}{k-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77cc12be5f995d37e9b238ed1dfc13ede59b230a)
- Vandermonderen identitatea. Izan bitez
, ![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8be5c6b5f3b0b234759a028c72755553e549783)
![{\displaystyle {\binom {2n}{n}}=\sum _{k=0}^{n}{\binom {n}{k}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/228e18e09b46a60da1013926947adee734ad5a48)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd16f34d5d3205f02b589e044d2afe0eb61b39ca)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/394ce156100e608039f4177369ed7b0a8725c9d3)
Ikus, gainera
- Konbinazio (konbinatoria)
Kanpo estekak