702 CHAPTER 12 Sequences and Series
Although it is possible to use Pascal’s triangle to find the coefficients in
for any positive integer value of n, it is impractical for large values of n. A more effi-
cient way to determine these coefficients uses the symbol n!(read“nfactorial”), de-
fined as follows.
1 x+ y 2 n
nFactorial (n!)For any positive integer n,
By definition, 0!1.
n!n 1 n 121 n 221 n 32 Á 122112.
Evaluating FactorialsEvaluate each factorial.
(a) (b)
(c) 0! is defined to be 1.
FIGURE 3shows how a graphing calculator computes factorials. NOW TRY
Evaluating Expressions Involving FactorialsFind the value of each expression.
(a)
(b)
(c)
(d) NOW TRY
Now look again at the coefficients of the expansion
The coefficient of the second term is 5, and the exponents on the variables in that
term are 4 and 1. From Example 2(a), The coefficient of the third term is
10, and the exponents are 3 and 2. From Example 2(b), Similar results are
true for the remaining terms. The first term can be written as and the last term
can be written as Then the coefficient of the first term should be and
the coefficient of the last term would be
The coefficient of a term in in which the variable part is is
This is called a binomial coefficient.The binomial coefficient is often represented by the symbol This nota-
tion comes from the fact that if we choose combinationsof nthings taken rat a time,
the result is given by that expression. We read as “combinations ofnthings
taken rat a time.”Another common representation is A.
nrB
nCr
nCr^.
n!
r! 1 n-r 2!n!
r! 1 n- r 2!
.
1 x+y 2 n xryn-r
5!0!5!= 1.
5!1 x 5!0!=1,
(^0) y (^5).
1 x^5 y^0 ,
5!3!2!=10.
5!4!1!=5.
1 x+y 25 = x^5 + 5 x^4 y+ 10 x^3 y^2 + 10 x^2 y^3 + 5 xy^4 + y^5.
4!
4!0!
=
4 # 3 # 2 # 1
14 # 3 # 2 # 12112= 1
6!
3!3!
=
6 # 5 # 4 # 3 # 2 # 1
13 # 2 # 1213 # 2 # 12=
6 # 5 # 4
3 # 2 # 1= 20
5!
3!2!
=
5 # 4 # 3 # 2 # 1
13 # 2 # 1212 # 12=
5 # 4
2 # 1= 10
5!
4!1!
=
5 # 4 # 3 # 2 # 1
14 # 3 # 2 # 12112= 5
EXAMPLE 2
0!= 1
3!= 3 # 2 # 1 = 6 5!= 5 # 4 # 3 # 2 # 1 = 120
EXAMPLE 1
FIGURE 3NOW TRY
EXERCISE 1
Evaluate.
7!
NOW TRY ANSWERS
- 5040
- (a) 28 (b) 56 (c) 1 (d) 6
NOW TRY
EXERCISE 2
Find the value of each
expression.
(a) (b)
(c) (d)