Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.

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2 Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads? 3 How many ways can you get exactly 3 heads?

3 Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? C(5, 1) = 5 2 How many ways can you get exactly 2 heads? 3 How many ways can you get exactly 3 heads?

4 Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? C(5, 1) = 5 2 How many ways can you get exactly 2 heads? C(5, 2) = 10 3 How many ways can you get exactly 3 heads?

5 Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? C(5, 1) = 5 2 How many ways can you get exactly 2 heads? C(5, 2) = 10 3 How many ways can you get exactly 3 heads? C(5, 3) = 10

6 Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? C(5, 1) = 5 2 How many ways can you get exactly 2 heads? C(5, 2) = 10 3 How many ways can you get exactly 3 heads? C(5, 3) = 10 and a more complicated question... 4 How many ways are there to get at most 2 heads?

7 4 How many ways are there to get at most 2 heads? Solution At most 2 heads means 0 heads or 1 head or 2 heads. 1 sequence has 0 heads. 5 sequences have 1 head. 10 sequences have 2 heads. So the number of sequences with at most 2 heads is = 16.

8 At least and at most Remember, at least at most For example, at least 5 means 5 or 6 or 7 or 8 or... For example, at most 5 means 5 or 4 or 3 or 2 or 1 or 0.

9 The Additive Principle The Additive Principle If you can choose one of m options OR one of n options, the total number of possibilities is m + n. Compare this with the Multiplicative Principle from before: The Multiplicative Principle If you have to choose one of m options AND one of n options, the total number of possibilities is mn.

10 The Additive Principle The Additive Principle If you can choose one of m options OR one of n options, the total number of possibilities is m + n. Compare this with the Multiplicative Principle from before: The Multiplicative Principle If you have to choose one of m options AND one of n options, the total number of possibilities is mn.

11 An important table or + and

12 Another question You flip a coin 5 times and record the sequence of heads and tails, just as before. 5 How many ways are there to get at least 2 heads?

13 The Complement Principle Instead of counting the number of good ways, sometimes it s easier to count the number of bad ways and subtract. The Complement Principle If you are trying to count the number of ways to do something in some good way, (# good ways) = (total # ways) (# bad ways)

14 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads?

15 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads? Bad solution: C(20, 2) + C(20, 3) + C(20, 4) + C(20, 5) + C(20, 6) +

16 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads? Bad solution: C(20, 2) + C(20, 3) + C(20, 4) + C(20, 5) + C(20, 6) + Good solution: If a good sequence has at least 2 heads, then a bad sequence has less than 2 heads.

17 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads? Bad solution: C(20, 2) + C(20, 3) + C(20, 4) + C(20, 5) + C(20, 6) + Good solution: If a good sequence has at least 2 heads, then a bad sequence has less than 2 heads. Total # of sequences: 2 20 = 1,048,576

18 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads? Bad solution: C(20, 2) + C(20, 3) + C(20, 4) + C(20, 5) + C(20, 6) + Good solution: If a good sequence has at least 2 heads, then a bad sequence has less than 2 heads. Total # of sequences: 2 20 = 1,048,576 # of bad sequences: C(20, 0) + C(20, 1) = 21

19 A bigger question You flip a coin 20 times and record the sequence of heads and tails, just as before. 6 How many ways are there to get at least 2 heads? Bad solution: C(20, 2) + C(20, 3) + C(20, 4) + C(20, 5) + C(20, 6) + Good solution: If a good sequence has at least 2 heads, then a bad sequence has less than 2 heads. Total # of sequences: 2 20 = 1,048,576 # of bad sequences: C(20, 0) + C(20, 1) = 21 # of good sequences: 1,048,555

20 Sock Questions You have 5 blue socks and 3 white socks in your sock drawer at random. You want to draw 3 socks from the drawer. 1 How many ways are there to do this? 2 How many ways can you do this and get 3 white socks? 3 How many ways can you do this and get 2 white socks and 1 blue sock? 4 How many ways can you do this and get 1 white sock and 2 blue socks? 5 How many ways can you do this and get 3 blue socks? 6 How many ways can you do this and get at least 2 white socks?

21 Multiple Steps Multi-Step Strategy To do a complicated problem, try to break it up into a sequence of smaller choices. Then we ll use the multiplicative principle to combine those smaller numbers.

22 A Short Summary We ve learned several principles for attacking these counting problems. Multi-Step Strategy: Break the problem up into a sequence of smaller choices. (Once we figure out those smaller numbers, we can multiply them together.) Additive Principle: If you can do X or Y, count the number of ways of each of them and add. Change the words at least and at most into several options with or. For example, at most 3 means 0 or 1 or 2 or 3. Complement Principle: If counting the good ways seems hard, maybe it would be easier to count the bad ways and subtract.

23 Practice Problems 1 How many 5-card poker hands have exactly 3 spades? 2 How many 5-card poker hands have at least one spade?

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