1. Use Combinations and the Binomial Theorem
Section 10.2
Use Combinations and the Binomial Theorem

2. California Standard:
20.0: Students know the binomial theorem and use it to expand the binomial expressions that are raised to positive integer exponents.

3. By following instructions, students will be able to:
OBJECTIVE(S):
By following instructions, students will be able to:
Use the combinations and the binomial theorem.

4. KEY CONCEPT Combinations (when ORDER is NOT IMPORTANT) Formula:
The number of combinations of r objects taken from a group of n distinct objects is denoted by
Formula:

5. EXAMPLE 1:
A standard deck of 52 playing cards has 4 suits with 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? 5

6. MULTIPLE EVENTS OCCURRING
A and B occurring at the same time.
Multiply
A or B can occur. Add

7. EXAMPLE 2:
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? 7

8. EXAMPLE 3:
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? 8

9. U-TRY #1:
Find the number of combinations. a) b) c) d)
e) In example 2, how many different sets of exactly 3 tragedies and 2 histories can you read?

10. PASCAL’S TRIANGLE
As numbers
As combinations

11. PASCAL’S TRIANGLE Pascal’s Triangle
Why is Pascal’s Triangle important? Pascal’s triangle is used to find the probability of an event occurring. Ex. In a 1 coin toss, what is the probability of getting 1 tail. H, T Row 1 = 1+1= 2 possible outcomes Choose 1 event = 1 Probability = ½ Ex. In a 3 coin toss, what is the probability of getting 2 tails? TTT, TTH, THH, THT, HTT, HHH, HTT, HTH Row 3 = = 8 possible outcomes Choose 2 event= 3 Probability = 3/8
Pascal’s Triangle

12. PASCAL’S TRIANGLE
Why is Pascal’s Triangle important? Pascal’s triangle can be used to find combinations. Ex. You have 4 pairs of jeans (B, R, W, P), how many different ways could you choose 3 pairs of jeans? BRW, RWP, BWP, BRP = 4 combinations
Pascal’s Triangle

13. EXAMPLE 4:
There are 6 Algebra students in the class. 2 Algebra students must join the math club to represent the Algebra group. Use Pascal’s triangle to find the number of combinations of 2 students that can be chosen to represent the group. 13

14. U-TRY #2:
In example 4, use Pascal’s triangle to find the number of combinations of 2 students that can be chosen if the Algebra class has 9 students.

15. BINOMIAL THEOREM

16. EXAMPLE 5: Use the binomial theorem to write the binomial expression.16

17. EXAMPLE 6: Use the binomial theorem to write the binomial expression.17

18. U-TRY #3: Use the binomial theorem to write the binomial expansion. a)c) d)

19. EXAMPLE 7:
Find the coefficient of in the expansion of . 19

20. HOMEWORK
Sec 10.2 WS