7.3 Finite series 199
The coefficient ofx
rin the expansion (7.11) is
(7.12)
and is called a binomial coefficient(sometimes read as ‘n choose r’). The binomial
coefficients are important in probability theory, where they are often given the symbol
n
C
r(orC
rn); we shall see in Chapter 21 that is the number of combinations of n
objects taken rat a time. In terms of binomial coefficients, the expansion is
(7.13)
A more general form of the binomial expansion is
(7.14)
EXAMPLE 7.4Calculate the binomial coefficients , and use them to expand
(i)(1 1 + 1 x)
5and (ii)(x 1 + 1 3)
5in powers of x.
By the definition of the binomial coefficients,
Therefore, remembering that 0! 1 = 11 ,
(i) By equation (7.13),
=+ +1 5 10 10 5+ + +
2345xx xxx
()1
5
0
5
1
5
2
501+=
xxxxxxxx
23455
3
5
4
5
5
5
0
5
5
5
05
1
5
1
5
=
=
!
!!
=,
=
44
5
14
5
5
2
5
3
5
=
!
!!
=,
=
=
!!
!!
=
23
10
55
5
5
5
012
rr
rr
r
=
−
=
!
!−!
,=,,,
()
..., 5
5
r
()xy
n
r
xy
nrnrnr+=
=−∑
0()1
0+=
=∑
x
n
r
x
nrnrn
r
n
r
nn n ...n r
r
n
rn r
=
−− −+
!
=
!
!−!
()( )( )
()
12 1