12.2 Combinations and Binomial Theorem

1. 12.2 Combinations and Binomial Theorem p. 708 What is a combination? How is it different from a permutation? What is the formula for nCr? What is Pascal’s Triangle? What is the Binomial Theorem? How is Pascal’s Triangle connected to the binomial theorem?

2. In the last section we learned counting problems where order was important • For other counting problems where order is NOT important like cards, (the order you’re dealt is not important, after you get them, reordering them doesn’t change your hand) • These unordered groupings are called Combinations

3. A Combination is a selection of “r” objects from a group of “n” objects where order is not important

4. Combination 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 nCr and is:

5. For instance, the number of combinations of 2 objects taken from a group of 5 objects is 2

6. Finding Combinations • In a standard deck of 52 cards there are 4 suits with 13 of each suit. • If the order isn’t important how many different 5-card hands are possible? • The number of ways to draw 5 cards from 52 is = 2,598,960

7. In how many of these hands are all 5 cards the same suit? • You need to choose 1 of the 4 suits and then 5 of the 13 cards in the suit. • The number of possible hands are:

8. How many 7 card hands are possible? • How many of these hands have all 7 cards the same suit?

9. When finding the number of ways both an event A and an event B can occur, you multiply. • When finding the number of ways that an event A ORB can occur, you +.

10. Deciding to + or * • A restaurant serves omelets. They offer 6 vegetarian ingredients and 4 meat ingredients. • You want exactly 2 veg. ingredients and 1 meat. How many kinds of omelets can you order?

11. Suppose you can afford at most 3 ingredients • How many different types can you order? • You can order an omelet with 0, or 1, or 2, or 3 items and there are 10 items to choose from.

12. Counting problems that involve ‘at least’ or ‘at most’ sometimes are easier to solve by subtracting possibilities you don’t want from the total number of possibilities.

13. Subtracting instead of adding: • A theatre is having 12 plays. You want to attend at least 3. How many combinations of plays can you attend? • You want to attend 3 or 4 or 5 or … or 12. • From this section you would solve the problem using: • Or……

14. For each play you can attend you can go or not go. • So, like section 12.1 it would be 2*2*2*2*2*2*2*2*2*2*2*2 =212 • And you will not attend 0, or 1, or 2. • So:

15. The Binomial Theorem • 0C0 • 1C0 1C1 • 2C02C12C2 • 3C03C13C2 3C3 • 4C04C14C24C34C4 • Etc…

16. Pascal's Triangle! • 1 • 1 1 • 1 2 1 • 1 3 3 1 • 1 4 6 4 1 • 1 5 10 10 5 1 • Etc… • This describes the coefficients in the expansion of the binomial (a+b)n

17. (a+b)2 = a2 + 2ab + b2 (1 2 1) • (a+b)3 = a3(b0)+3a2b1+3a1b2+b3(a0) (1 3 3 1) • (a+b)4 = a4+4a3b+6a2b2+4ab3+b4 (1 4 6 4 1) • In general…

18. (a+b)n (n is a positive integer)= • nC0anb0 + nC1an-1b1 + nC2an-2b2 + …+ nCna0bn • =

19. (a+3)5 = • 5C0a530+5C1a431+5C2a332+5C3a233+5C4a134+5C5a035= • 1a5 + 15a4 + 90a3 + 270a2 + 405a + 243

20. (a+3)5 = • 1a530+5a431+10a332+10a233+5a134+1a035 • 1a5 + 15a4 + 90a3 + 270a2 + 405a + 243 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

21. What is a combination? A selection of objects where order is not important. • How is it different from a permutation? In other counting problems, order is not important. • What is the formula for nCr? • What is Pascal’s Triangle? An arrangement of the values of nCr in a triangular pattern. • What is the Binomial Theorem? A formula that gives the coefficients in the raising of (a+b) to a power. • How is Pascal’s Triangle connected to the binomial theorem? Pascal’s Triangle gives the same coefficients as the Binomial Theorem.

22. Assignment 12.2 • P. 712 19-39 odd, 45-57 odd, skip 51