- Source: Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
(
a
+
b
+
c
)
n
=
∑
i
,
j
,
k
i
+
j
+
k
=
n
(
n
i
,
j
,
k
)
a
i
b
j
c
k
,
{\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}
where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. The trinomial coefficients are given by
(
n
i
,
j
,
k
)
=
n
!
i
!
j
!
k
!
.
{\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.}
This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.
Derivation
The trinomial expansion can be calculated by applying the binomial expansion twice, setting
d
=
b
+
c
{\displaystyle d=b+c}
, which leads to
(
a
+
b
+
c
)
n
=
(
a
+
d
)
n
=
∑
r
=
0
n
(
n
r
)
a
n
−
r
d
r
=
∑
r
=
0
n
(
n
r
)
a
n
−
r
(
b
+
c
)
r
=
∑
r
=
0
n
(
n
r
)
a
n
−
r
∑
s
=
0
r
(
r
s
)
b
r
−
s
c
s
.
{\displaystyle {\begin{aligned}(a+b+c)^{n}&=(a+d)^{n}=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,d^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,(b+c)^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,\sum _{s=0}^{r}{r \choose s}\,b^{r-s}\,c^{s}.\end{aligned}}}
Above, the resulting
(
b
+
c
)
r
{\displaystyle (b+c)^{r}}
in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index
s
{\displaystyle s}
.
The product of the two binomial coefficients is simplified by shortening
r
!
{\displaystyle r!}
,
(
n
r
)
(
r
s
)
=
n
!
r
!
(
n
−
r
)
!
r
!
s
!
(
r
−
s
)
!
=
n
!
(
n
−
r
)
!
(
r
−
s
)
!
s
!
,
{\displaystyle {n \choose r}\,{r \choose s}={\frac {n!}{r!(n-r)!}}{\frac {r!}{s!(r-s)!}}={\frac {n!}{(n-r)!(r-s)!s!}},}
and comparing the index combinations here with the ones in the exponents, they can be relabelled to
i
=
n
−
r
,
j
=
r
−
s
,
k
=
s
{\displaystyle i=n-r,~j=r-s,~k=s}
, which provides the expression given in the first paragraph.
Properties
The number of terms of an expanded trinomial is the triangular number
t
n
+
1
=
(
n
+
2
)
(
n
+
1
)
2
,
{\displaystyle t_{n+1}={\frac {(n+2)(n+1)}{2}},}
where n is the exponent to which the trinomial is raised.
Example
An example of a trinomial expansion with
n
=
2
{\displaystyle n=2}
is :
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
a
b
+
2
b
c
+
2
c
a
{\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca}
See also
Binomial expansion
Pascal's pyramid
Multinomial coefficient
Trinomial triangle
References
Kata Kunci Pencarian:
- Trinomial expansion
- Pascal's pyramid
- Multinomial theorem
- Trinomial
- List of factorial and binomial topics
- Pascal's triangle
- Trinomial coefficient
- Binomial coefficient
- Hardy–Weinberg principle
- Trinomial triangle
