Evaluating Binomial CoefficientsEvaluate each binomial coefficient.
(a) Let.
Subtract.Definition of nfactorialLowest termsBinomial coefficients will always be whole numbers.
(b)
(c)
FIGURE 4shows how a graphing calculator displays the binomial coefficients computed
here. NOW TRY
We now state the binomial theorem,or the general binomial expansion.
6 C 3 =
6!
3! 16 - 32!
=
6!
3!3!
=
6 # 5 # 4 # 3 # 2 # 1
3 # 2 # 1 # 3 # 2 # 1= 20
5 C 3 =
5!
3! 15 - 32!
=
5!
3!2!
=
5 # 4 # 3 # 2 # 1
3 # 2 # 1 # 2 # 1= 10
= 5
=
5 # 4 # 3 # 2 # 1
4 # 3 # 2 # 1 # 1=
5!
4!1!
5 C 4 = n=5, r=^4
5!
4! 15 - 42!
EXAMPLE 3
Formula for the Binomial CoefficientFor nonnegative integers nand r, where
nCr
n!
r! 1 nr 2!
.
r...n,
nCrBinomial TheoremFor any positive integer n,
n!
3! 1 n 32!
xn^3 y^3 ###
n!
1 n 12 !1!
xyn^1 yn.
1 xy 2 nxn
n!
1! 1 n 12!
xn^1 y
n!
2! 1 n 22!
xn^2 y^2
The binomial theorem can be written in summation notation as
NOTE We used the letter kas the summation index letter in the statement just given.
This is customary notation in mathematics.
1 xy 2 n a
nk 0n!
k! 1 nk 2!
xnk^ yk.
FIGURE 4NOW TRY
EXERCISE 3
Evaluate. 7 C 2
NOW TRY ANSWER
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