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Paskal üçburçlugy
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Paskal üçburçlugyň ilkinji 15 setiri (
n
= 0, 1, …, 14)
Paskal üçburçlugy
—Binomial koeffisientler we Paskal üçburçlugy şeýle bolsa,
(
x
+
y
)
n
=
∑
k
=
0
n
(
n
k
)
x
n
−
k
y
k
{\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}}
onda
(
n
k
)
=
(
n
−
1
k
−
1
)
+
(
n
−
1
k
)
{\displaystyle {n \choose k}={n-1 \choose k-1}+{n-1 \choose k}}
bolar.
Formula
(
n
k
)
=
n
!
k
!
⋅
(
n
−
k
)
!
=
n
1
⋅
n
−
1
2
⋅
…
⋅
n
−
k
+
1
k
{\displaystyle {n \choose k}={\frac {n!}{k!\cdot (n-k)!}}={\frac {n}{1}}\cdot {\frac {n-1}{2}}\cdot \ldots \cdot {\frac {n-k+1}{k}}}
,
n
,
k
∈
N
0
{\displaystyle n,k\in \mathbb {N} _{0}}
bolsa
Meselem
(
5
3
)
=
5
!
3
!
⋅
2
!
=
5
⋅
4
⋅
3
!
2
!
⋅
3
!
=
5
⋅
4
2
!
=
20
2
=
10
{\displaystyle {5 \choose 3}={\frac {5!}{3!\cdot 2!}}={\frac {5\cdot 4\cdot 3!}{2!\cdot 3!}}={\frac {5\cdot 4}{2!}}={\frac {20}{2}}=10}