1. WARMUP Lesson 10.2, For use with pages 690-69b
Evaluate the expression.
1. 5!
ANSWER
120
2. (4 – 2)!3!
ANSWER 12
3. How do you say “!” in math? What does it mean?
ANSWER
Factorial. It means to multiply that number by ALL the Natural numbers less than it.
4. 7 P5
ANSWER
2520
5. In how many ways can 6 people line up to buy tickets for a movie?
ANSWER
720

2. 10.2 Notes – Combinations and the Binomial Theorem

3. Objective -To use combinations to count the number of ways an event can happen. To use the binomial theorem to expand binomials.
A combination is a selection of objects from a group where order is not important.

4. Using a standard deck of 52 cards how many 4-card hands
are possible?
From a standard deck, how many 4 card hands have the same suit?
First: You need to choose 1 of the 4 suits.
Then: Choose 4 out of the 13 cards in the suit.

5. You can order a sandwich with 1, 2, 3 or 4 different kinds of
You are ordering a lunch and can pick from 5 main dishes, 7 side dishes and 6 drinks? How many combinations of meals
are possible if you pick 2 main dishes, 3 side dishes and 2 drinks?
Are
Choose M 5 2 S 7 3 D 6
You can order a sandwich with 1, 2, 3 or 4 different kinds of
meats. If there are 8 meats to pick from, how many possible
sandwiches are there?

6. If you wanted to visit at least 9 of the 13 parks and resorts,
You are taking a trip. You can pick from 8 amusement parks and 4 beach resorts. Suppose you want to visit 5 amusement parks and 2 beach resorts. How many different trips are possible?
Are
Choose AP 8 5 BR 4 2
If you wanted to visit at least 9 of the 13 parks and resorts,
how many different kinds of trips could you go on?

7. A movie theater is showing 8 different movies
A movie theater is showing 8 different movies. You would like to see at least 5 of the movies. How many different combinations of movies can you see?
A ice cream shop has 9 flavors of ice cream to pick from. You would like to pick at least 3 flavors. How many combinations of can you pick?

8. Pascal’s Triangle

9. Each number is the sum of the two numbers
Pascal’s Triangle
Each number is the sum of the two numbers
directly above it.

10. Sum of each row are powers of 2.
Pascal’s Triangle
Sum of each row 20 21 22 23 24 25
Sum of each row are powers of 2.

11. Pascal’s Triangle can be used to expand binomials.
Third row of Pascal’s Triangle

12. Expand: (a + b)2 = = (a + b)(a + b)
= a2 + ab + ba + b2 = 1a2 + 2ab + 1b2
(a + b)3 =
= (a + b)(a + b)(a + b)
= (a2 + 2ab + b2)(a + b)
= a3 + 2a2b + b2a + ba2 + 2ab2 + b3
= 1a3 + 3a2b + 3b2a + 1b3
3. (a + b)4 =

13.
a a a a a
b b b b b
Steps:
1. Write in your coefficients from Pascal’s Triangle (red)
2. Write in your two “things” (a and b here)
3. Put in your exponents counting down to ZERO
And vice Versa
If it was a MINUS sign, put in PLUS, MINUS, PLUS…

14. Expand: (a + 7)2 = = (a + 7)(a + 7)
= a2 + 7a + 7a + 49 = a2 + 14a + 49
(a + 7)3 =
= (a + 7)(a + 7)(a + 7)
= (a2 + 14a + 49)(a + 7)
= a3 + 14a2 + 49a + 7a2 + 98a + 343
= a3 + 21a2 + 14ba + 343
3. (a + 7)4 =

15.
a a a a a

16.
x x x x
(-3a) (-3a) (-3a) (-3a)

17. 1st Thing
2nd: 5 – 1 = 4
3rd: Same Number
4th: 8 – 4 = 4

18. “Hints”
0! = 1
An Ace is not a Face.

19. Scene from the Brad Pitt Movie: Moneyball

20. EXAMPLE 1
Find combinations
CARDS
A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit.
If the order in which the cards are dealt is not important, how many different 5-card hands are possible?
In how many 5-card hands are all 5 cards of the same color?

21. EXAMPLE 1
Find combinations
SOLUTION
The number of ways to choose 5 cards from a deck of 52 cards is:
4b! 5! =
52!
52C5 = b!
4b! 5!
= 2,598,960

22. EXAMPLE 1
Find combinations
For all 5 cards to be the same color, you need to choose 1 of the 2 colors and then 5 of the 26 cards in that color. So, the number of possible hands is:
21! 5! =
26!
26C5
2C1
1! ! 2! = !
21! 5! 2
= 131,560

23. EXAMPLE 2
Decide to multiply or add combinations
THEATER
William Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies.
How many different sets of exactly 2 comedies and 1 tragedy can you read?
How many different sets of at most 3 plays can you read?

24. EXAMPLE 2
Decide to multiply or add combinations
SOLUTION
You can choose 2 of the 18 comedies and 1 of the 10 tragedies. So, the number of possible sets of plays is:
9! 1! =
10!
10C1
18C2
16! !
18! =
b !
16! 9! ! = =

25. EXAMPLE 2
Decide to multiply or add combinations
You can read 0, 1, 2, or 3 plays. Because there are 38 plays that can be chosen, the number of possible sets of plays is:
38C0 + 38C1 + 38C2 +38C3
= b
= 91b8

26. EXAMPLE 3
Solve a multi-step problem
BASKETBALL
During the school year, the girl’s basketball team is scheduled to play 12 home games. You want to attend at least 3 of the games. How many different combinations of games can you attend?
SOLUTION
Of the 12 home games, you want to attend 3 games, or 4 games, or 5 games, and so on. So, the number of combinations of games you can attend is:
12C3 + 12C4 + 12C5 +…+ 12C12

27. EXAMPLE 3
Solve a multi-step problem
Instead of adding these combinations, use the following reasoning. For each of the 12 games, you can choose to attend or not attend the game, so there are 212 total combinations. If you attend at least 3 games, you do not attend only a total of 0, 1, or 2 games. So, the number of ways you can attend at least 3 games is:
212 –
(12C0 + 12C1 + 12C2 )
= – ( )
= 401b

28. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the number of combinations.
8C3 1. 56
ANSWER

29. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the number of combinations. 2.
10C6
210
ANSWER

30. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the number of combinations.
bC2 3. 21
ANSWER

31. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the number of combinations. 4.
14C5
2002
ANSWER

32. GUIDED PRACTICE
for Examples 1, 2 and 3
WHAT IF? In Example 2, how many different sets of exactly 3 tragedies and 2 histories can you read?
5400 sets
ANSWER

33. EXAMPLE 4
Use Pascal’s triangle
School Clubs
The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives.
SOLUTION
Because you need to find 6C2, write the 6th row of Pascal’s triangle by adding numbers from the previous row.

34. EXAMPLE 4
Use Pascal’s triangle
n = 5 (5th row)
n = 6 (6th row)
6C0
6C1
6C2
6C3
6C4
6C5
6C6
ANSWER
The value of 6C2 is the third number in the 6th row of Pascal’s triangle, as shown above. Therefore, 6C2 = 15. There are 15 combinations of representatives for the convention.

35. GUIDED PRACTICE
for Example 4 6.
WHAT IF? In Example 4, use Pascal’s triangle to find the number of combinations of 2 members that can be chosen if the Model UN club has b members.
ANSWER
21 combinations

36. EXAMPLE 5
Expand a power of a binomial sum
Use the binomial theorem to write the binomial expansion.
(x2 + y)3
= 3C0(x2)3y0 + 3C1(x2)2y1 + 3C2(x2)1y2 + 3C3(x2)0y3
= (1)(x6)(1) + (3)(x4)(y) + (3)(x2)(y2) + (1)(1)(y3)
= x6 + 3x4y + 3x2y2 + y3

37. EXAMPLE 6
Expand a power of a binomial difference
Use the binomial theorem to write the binomial expansion.
(a – 2b)4
= [a + (–2b)]4
= 4C0a4(–2b)0 + 4C1a3(–2b)1 + 4C2a2(–2b)2 + 4C3a1(–2b)3
+ 4C4a0(–2b)4
= (1)(a4)(1) + (4)(a3)(–2b) + (6)(a2)(4b2) + (4)(a)(–8b3)
+ (1)(1)(16b4)
= a4 – 8a3b + 24a2b2 – 32ab3 + 16b4

38. GUIDED PRACTICE
for Examples 5 and 6
Use the binomial theorem to write the binomial expansion.
(x + 3)5
x5 + 15x4+ 90x3 + 2b0x x + 243
ANSWER

39. GUIDED PRACTICE
for Examples 5 and 6
Use the binomial theorem to write the binomial expansion.
(a + 2b)4
a4 + 8a3b + 24a2b2 + 32ab3 + 16b4
ANSWER

40. GUIDED PRACTICE
for Examples 5 and 6
Use the binomial theorem to write the binomial expansion.
(2p – q)4
16p4 – 32p3q + 24p2q2 – 8pq3 + q4
ANSWER

41. GUIDED PRACTICE
for Examples 5 and 6
Use the binomial theorem to write the binomial expansion.
(5 – 2y)3
–8y3 + 60y2 – 150y + 125
ANSWER

42. EXAMPLE b
Find a coefficient in an expansion
Find the coefficient of x4 in the expansion of (3x + 2)10.
SOLUTION
From the binomial theorem, you know the following:
(3x + 2)10
= 10C0(3x)10(2)0 + 10C1(3x)9(2) C10(3x)0(2)10
Each term in the expansion has the form 10Cr(3x)10 – r (2) r. The term containing x4 occurs when r = 6:
10C6(3x)4(2)6
= (210)(81x4)(64)
= 1,088,640x4
ANSWER
The coefficient of x4 is 1,088,640.

43. GUIDED PRACTICE
for Example b
Find the coefficient of x5 in the expansion of (x – 3)b.
11.
ANSWER
189

44. GUIDED PRACTICE
for Example b
12.
Find the coefficient of x3 in the expansion of (2x + 5)8.
ANSWER
1,400,000

45. 10.2 Assignment
10.2: ODD, ODD (use the way we did it in class), 49, 54-57