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Advanced Algebra Notes
Section 10.2 Use Combination and the Binomial Theorem
You learned in section 10.1 that order was important for some counting problems.
Today we learn about the selection of r objects from a group of n objects were the order
combination
is not important, and this is called a _______________.
Combinations Of n Objects Taken r At A Time
The number of combinations of r objects taken from a group of n distinct
objects is denoted by:
nC r 
Example 1:
n
 n  r ! r !
You are picking 7 books from a stack of 32. If the order of the books you
choose is not important, How many different 7 book groups are possible?
32!
25!7!
1.696391424 E 10

5040
 3,365,856
32 C 7 
When finding the number of ways both an Event A and an Event B can occur, you need
multiply
to ___________.
Example 2:
In how many ways can you choose 3 red cards and 2 black cards from a
standard deck of cards.
26 C 3 
26 C 2 

26!
26!

23! 3! 24!2!
26  25  24 26  25

3  2 1
2 1
 845, 000
When finding the number of ways an Event A or an Event B can occur, you need to
add
______.
Example 3:
The local movie rental store is having a special on new releases. The new
releases consist of 12 comedies, 8 action, 7 drama, 5 suspense, and 9 family
movies. You can afford at most 2 movies. How many movie combinations
can you rent?
41!
41!
41!
41
C0 
41
C1 
41
C 2

41! 0! 40!1!
 1  41  820
 862

39! 2!
Counting problems that involve phrases like “at least” or “ at most” are sometimes easier
subtracting possibilities you do not want from the total number of
to solve by ____________
possibilities.
Example 4:
A popular magazine has 11 articles. You want to read at least 2 of the
articles. How many different combinations of articles can you read?
211  (11 C 0 
11
C 1)  2048  (1  11)
 2048 12
 2036
If we would arrange the values of nCr in a triangular pattern in which each row
corresponds to a value n, you get what is called _________________.
Pascal’s Triangle
Pascal’s Triangle is named after the French mathematician Blaise Pascal (1623-1662).
Pascal’s Triangle
Row
n=0
n=1
n=2
n=3
n=4
n=5
Combinations
0 C0
1 C0
1 C1
2 C0
2 C1
2 C2
3C0
3C1
3C2
3C3
4 C0
4 C1
4 C2
4 C3
4 C4
5 C0
5 C1
5 C2
5 C3
5 C4
5 C5
Numbers
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Example 5: Use Pascal’s triangle to find the number of combinations.
Out of 5 finalists, your class must choose 3 class representatives.
Find
5
C3
Using the 5th row of Pascal’s Triangle
10 combinations of class representatives
There is an important relationship between powers of binomials and combinations. The
numbers in Pascal’s Triangle can be used to find coefficients in binomial expansions.
Example: The coefficients in the expansion of (a + b)4 are the numbers of combinations
4
in the row of Pascal’s triangle for n = ______.
(a + b)4 =
4C0 a
4 b0
+ 4C1 a3b + 4C2 a2b2 + 4C3 ab3 + 4C4 b4
1 a4 + ____
4 a3b + ____
6 a2b2 + ____
4 ab3 + ____
1 b4
= ____
Example 6: Use the binomial theorem to write the binomial expansion.
A)
(x + 4)3
3
0
2
1
2
0
3
C
x
(4)

C
x
(4)

C
x
(4)

C
x
(4)
3
0
3 1
3
2
3
3
1x3 (1)  (3) x 2 (4)  3 x(16) 1(1)(64)
x3  12 x2  48 x  64
To expand a power of a binomial difference, you can write the binomial as a sum. The
alternate between ______
+ and
resulting expansion will have terms whose signs ____________
______.
B)
(2m – n)4
4
0
3
1
2
2
1
3
0
4
C
(2
m
)
(

n
)

C
(2
m
)
(

n
)

C
(2
m
)
(

n
)

C
(2
m
)
(

n
)

C
(2
m
)
(

n
)
4
0
4
1
4
2
4
3
4
4
1(16m4 )(1)  4(8m3 )(n)  6(4m2 )(n2 )  4(2m)( n3 ) 1(1)( n4 )
16m4  32m3 n  24m2 n2  8mn3  n4
Example 7: Find a coefficient in an expansion.
Find the coefficient of the term x5 in the expansion of (2x – 7)9.
Each term in the expansion has the form:
nr
r
C
(
a
)
(
b
)
, where n  exp onent in the exp ansion
n
r
r  exp onent in exp ansion  exp onent in term
5
4
C
(2
x
)
(

7)
9
4
 126(32 x5 )(2401)
 9,680,832x5
coefficient is 9,680,832