Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY ALGEBRA

In fact, the general expression, thebinomial expansionfor powern, is given by

(x+y)n=

∑k=n

k=0

nC

kx

n−kyk, (1.49)

wherenCkis called thebinomial coefficientand is expressed in terms of factorial

functions byn!/[k!(n−k)!]. Clearly, simply to make such a statement does not

constitute proof of its validity, but, as we will see in subsection 1.5.2, (1.49) can

beprovedusing a method called induction. Before turning to that proof, we

investigate some of the elementary properties of the binomial coefficients.

1.5.1 Binomial coefficients

As stated above, the binomial coefficients are defined by

nC

k≡

n!

k!(n−k)!

≡

(

n

k

)

for 0≤k≤n, (1.50)

where in the second identity we give a common alternative notation fornCk.

Obvious properties include

(i)nC 0 =nCn=1,

(ii)nC 1 =nCn− 1 =n,

(iii)nCk=nCn−k.

We note that, for any givenn, the largest coefficient in the binomial expansion is

the middle one (k=n/2) ifnis even; the middle two coefficients (k=^12 (n±1))

are equal largest ifnis odd. Somewhat less obvious is the result

nC

k+

nC

k− 1 =

n!

k!(n−k)!

+

n!

(k−1)!(n−k+1)!

=

n![(n+1−k)+k]

k!(n+1−k)!

=

(n+1)!

k!(n+1−k)!

=n+1Ck. (1.51)

An equivalent statement, in whichkhas been redefined ask+1, is

nC

k+

nC

k+1=

n+1C

k+1. (1.52)

1.5.2 Proof of the binomial expansion

We are now in a position toprovethe binomial expansion (1.49). In doing so, we

introduce the reader to a procedure applicable to certain types of problems and

known as themethod of induction. The method is discussed much more fully in

subsection 1.7.1.