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

2. Order of events or outcomes DOES MATTER!!!
Combination:
Order of events or outcomes IS NOT IMPORTANT!!!
Permutation:
Order of events or outcomes DOES MATTER!!!

3. Combinations
nCr
Examples:
8C3
10C6
7C2
14C5

4. Pascal’s Triangle!

5. Finding Combinations
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?
4 queens and 1 other card
5 non-face cards

6. Solutions:
The number of ways to choose 5 cards from a deck of 52 cards is:
52C5
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:
2C1 · 26C5

7. Multiple Events
When finding the number of ways both event A AND event B can occur, you MULTIPLY.
When finding the number of ways that event A OR event B can occur, you ADD.
At least one heart
At least one Ace

8. 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?

9. During the school year, the girls 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?

10. Binomial Expansion
There is a relationship between the powers of binomials and combinations.
Example:
4C C C C C4

11. Binomial Theorem:
Whenever a binomial has a POSITIVE EXPONENT (n is positive), the binomial expansion of (a + b)n is:
nC0anb0 + nC1an-1b1+nC2an-2b2+…+nCna0bn
This sequence stops at n (the power).

12. Use the binomial theorem to write the binomial expansion.

13. Use the binomial theorem to write the binomial expansion.

14. Use the binomial theorem to write the binomial expansion.

15. Use the binomial theorem to write the binomial expansion.
*** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

16. Use the binomial theorem to write the binomial expansion.
*** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

17. Use the binomial theorem to write the binomial expansion.
*** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

18. Homework:
P , 13-18, 24-34, 38-40