This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.

1. 11.4 Factorials and the
Binomial Theorem
Chapter 11 Further Topics in Algebra

2. Concepts and Objectives
 Factorials and the Binomial Theorem
 Be able to use the definition of factorials to simplify
expressions containing factorials, or to express in
factorial form expressions containing products of
consecutive integers.
 Given a binomial power, expand it as a binomial
series in one step
 Given a binomial power of the form , find term
number k, or find the term which contains br, where k
and r are integers from 0 through n.
 

n
a b

3. Factorials
 The expression n! (read “n factorial”) means the product
of the first n consecutive positive integers.
For example, 5! = 5  4  3  2  1 = 120
also, 5! = 5  4  3  2  1
= 5  4!
This behavior leads to a very important property:
 
 
! 1 !
n n n

4. Factorials (cont.)
 Just as we can multiply n–1! by n to produce n!, we can
reverse the process and divide n! by n to produce n–1! :
 Thus, 0! = 1.
4! 24
3! 6
2! 2
1! 1
0! 1
?
 1
 2
 3
 4

1
1
1

5. Factorials (cont.)
 Fractions which have factorials in the numerator and
denominator can often be cancelled.
 Example: Simplify
10!
7!

6. Factorials (cont.)
 Fractions which have factorials in the numerator and
denominator can often be cancelled.
 Example: Simplify
10!
7!

10! 10 9 8 7!
7! 7!
10 9 8
720

7. Factorials (cont.)
 When dealing with variables, keep the definition of a
factorial in mind.
 Example: Simplify
 
 


1 !
1 !
n
n

8. Factorials (cont.)
 When dealing with variables, keep the definition of a
factorial in mind.
 Example: Simplify
 
 


1 !
1 !
n
n
 
 
   
 
  

 
1 ! 1 1 !
1 ! 1 !
n n n n
n n
  
 1
n n

9. Binomial Series
 A binomial squared becomes
 A binomial cubed becomes
 
   
2 2 2
2
a b a ab b
    
   
3 2
a b a b a b
  
   
2 2
2
a b a ab b
     
2 2
3 3
2 2
2 2
a b a b ab
a b b
a
   
3 2 2 3
3 3
a a b ab b

10. Binomial Series (cont.)
 As you may recall from Algebra II, the coefficients
correspond to rows from Pascal’s Triangle
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1

11. Binomial Series (cont.)
 Example: Expand  

5
2 1
x

12. Binomial Series (cont.)
 Example: Expand
a = 2x and b = 1; the exponents begin and end at 5 (a
goes down while b goes up). Looking at row 5 on the
triangle, our coefficients are 1, 5, 10, 10, 5, 1, so we write
our expression as follows:
(Notice that the exponents apply to the entire term of the
binomial, not just the variable.)
 

5
2 1
x
                  
    
5 4 3 2 2 3 4 5
5 10 1
1 1 1
2 2 2 1
2 1
5 2
0
x x x x x
     
5 4 3 2
32 80 80 40 10 1
x x x x x
    
5 4 3 2 2 3 4 5
5 10 10 5
a a a a a
b b b b b

13. Binomial Series (cont.)
 Consider the binomial series :
If we multiply the coefficient of a term by a fraction
consisting of the exponent of a over the term number,
we get the coefficient of the next number.
 

7
a b
       
7 6 5 2 4 3 3 4 2 5 6 7
7 21 35 35 21 7
a a b a b a b a b a b ab b
8
7
6
5
4
3
2
1
   
 
   
   
exp. 7
coeff. 1 7,
term # 1
 

 
 
6
7 21,
2
 
 
 
5
21 =35, ...
3

14. Binomial Series (cont.)
 Now let’s see what happens to if we don’t
simplify the fractions as we calculate them:
 

8
a b
1
2
3
4
5
8
a
7
8
1
a b
6 2
8 7
1 2
a b
5 3
8 7 6
1 2 3
a b
4 4
8 7 6 5
1 2 3 4
a b
Do you see the pattern?
What is it?

15. Binomial Series (cont.)
 The coefficients of a binomial series can be written as
factorials, much as we did earlier. For example, let’s
look at the coefficient for the fourth term:

8 7 6 8 7 6
1 2 3 1 2 3

8 7 6 5!
1 2 3 5!

8!
3! 5!

16. Binomial Series (cont.)
 Looking back at the original expression:
Notice how the numbers in the coefficient expression
are found elsewhere in the expression.
 8 is the value of the exponent to which a + b is
raised.
 5 is the value of a’s exponent and 3 is the value of b’s.
 The exponent of b is always one less than the term
number (4).
 
   
5 3
8 !
... ...
! !
5
3
8
a b a b

17. Binomial Theorem
 The formula for the term containing br of a + bn,
therefore, is
or nCr
 Example: Find the term containing y6 of
 


!
! !
n r r
n
a b
r n r
n
r
 
  
 
 10
8
x y


18. Binomial Theorem (cont.)
 The formula for the term containing br of a + bn,
therefore, is
or nCr
 Example: Find the term containing y6 of
 


!
! !
n r r
n
a b
r n r
n
r
 
  
 
 10
8
x y

 
  
 
 
6
10 6 4 6
10! 10!
8 262144
6! 10 6 ! 6! 4!
x y x y

 

  4 6
10 9 8 7
262144
4 3 2 1
x y

3
4 6
55,050,240x y


19. Binomial Theorem (cont.)
 Example: Find the term in which contains f 23.
 

58
e f

20. Binomial Theorem (cont.)
 Example: Find the term in which contains f 23.
Since n = 58, n – r = 58 – 23 = 35. Therefore the term is
(When dealing with negative terms such as f, recall that
even exponents will produce positive terms and odd
exponents will produce negative terms.)
 

58
e f
 35 23
58!
35! 23!
e f

21. Binomial Theorem (cont.)
 Similarly, the kth term of binomial expansion of
is found by realizing that the exponent of b will be k – 1,
which gives us the formula:
(replace r with k – 1)
 n
a b

   
 
 
1 1
!
1 ! 1 !
n k k
n
a b
k n k
  

  
 
1 1
1
n k k
n
a b
k
  
 
 

 

22. Binomial Theorem (cont.)
 Example: Find the 4th term of  12
2c d


23. Binomial Theorem (cont.)
 Example: Find the 4th term of
n = 12, k = 4, which means that k – 1 = 3
 12
2c d

     
9 3 9 3
12! 10 11 12
2 512
3!9! 1 2 3
c d c d
  
5 4
9 3
112,640c d
 

24. Binomial Theorem (cont.)
 Desmos can also find the coefficient using a function
called nCr(n, r):
4th term of
n = 12, k – 1 = 3, n – k –1 = 9
 12
2c d

   
9 3
12
2
3
c d
 

 
 

25. Binomial Theorem Practice
 Example: Expand  

5
2 3
x

26. Binomial Theorem Practice
 Example: Expand  

5
2 3
x
         
   
   
   
   
5 4 3 2
5! 5!
2 2 3 2 3
1!4! 2!3!
x x x
        
   
     
   
   
2 3 4 5
5! 5!
2 3 2 3 3
3!2! 4!1!
x x
     
5 4 3 2
32 240 720 1080 810 243
x x x x x

27. Binomial Theorem Practice
 Example: Simplify ( )
 

6
1 i  1
i

28. Binomial Theorem Practice
 Example: Simplify ( )
 

6
1 i  1
i
        
     
  
     
     
6 5 4 2 3 3
6! 6! 6!
1 1 1 1
1!5! 2!4! 3!3!
i i i
     
   
  
   
   
2 4 5 6
6! 6!
1 1
4!2! 5!1!
i i i
      
2 3 4 5 6
1 6 15 20 15 6
i i i i i i
      
1 6 15 20 15 6 1
i i i
 8i

29. Binomial Theorem Practice
 Example: Expand
3
3
2 4
r r
 

 
 

30. Binomial Theorem Practice
 Example: Expand
3
3
2 4
r r
 

 
 
2 3
3 2
3 3 3
2 2 4 2 4 4
3 3
r r r
r r r
     
     
   
     
     
     
     
9 4
3
8 48 96 64
r r
r r r r
   
13 3
9 4
2 2
8 48 96 64
r r
r r
   
I will accept either

31. Classwork
 11.4 Assignment (College Algebra)
 Page 1034: 2-20 (even); page 1023: 22-38 (even;
omit 34); page 1015: 50-60 (even), 70, 72
 11.4 Classwork Check
 Quiz 11.3