Coin Flip Questions

Suppose you flip a coin five times and write down the sequenceof 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?

Coin Flip Questions

Suppose you flip a coin five times and write down the sequenceof results, like “HHHHH” or “HTTHT.”

1 How many ways can you get exactly 1 head? C(5,1) = 52 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?

Coin Flip Questions

Suppose you flip a coin five times and write down the sequenceof results, like “HHHHH” or “HTTHT.”

1 How many ways can you get exactly 1 head? C(5,1) = 52 How many ways can you get exactly 2 heads? C(5,2) = 103 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?

Coin Flip Questions

Suppose you flip a coin five times and write down the sequenceof results, like “HHHHH” or “HTTHT.”

1 How many ways can you get exactly 1 head? C(5,1) = 52 How many ways can you get exactly 2 heads? C(5,2) = 103 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?

Coin Flip Questions

Suppose you flip a coin five times and write down the sequenceof results, like “HHHHH” or “HTTHT.”

1 How many ways can you get exactly 1 head? C(5,1) = 52 How many ways can you get exactly 2 heads? C(5,2) = 103 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?

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 is1 + 5 + 10 = 16.

“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.

The Additive Principle

The Additive PrincipleIf you can choose one of m options OR one of n options, thetotal number of possibilities is

m + n.

Compare this with the Multiplicative Principle from before:

The Multiplicative PrincipleIf you have to choose one of m options AND one of n options,the total number of possibilities is

mn.

The Additive Principle

The Additive PrincipleIf you can choose one of m options OR one of n options, thetotal number of possibilities is

m + n.

Compare this with the Multiplicative Principle from before:

The Multiplicative PrincipleIf you have to choose one of m options AND one of n options,the total number of possibilities is

mn.

An important table

or +and ×

Another question

You flip a coin 5 times and record the sequence of heads andtails, just as before.

5 How many ways are there to get at least 2 heads?

The Complement Principle

Instead of counting the number of good ways, sometimes it’seasier to count the number of bad ways and subtract.

The Complement PrincipleIf you are trying to count the number of ways to do something insome “good” way,

(# good ways) = (total # ways)− (# bad ways)

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576

−

# of bad sequences: C(20,0) + C(20,1) = 21# of good sequences: 1,048,555

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576

−

# of bad sequences: C(20,0) + C(20,1) = 21# of good sequences: 1,048,555

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576

−

# of bad sequences: C(20,0) + C(20,1) = 21# of good sequences: 1,048,555

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576

−

# of bad sequences: C(20,0) + C(20,1) = 21# of good sequences: 1,048,555

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576

−

# of bad sequences: C(20,0) + C(20,1) = 21

# of good sequences: 1,048,555

A bigger question

You flip a coin 20 times and record the sequence of heads andtails, 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: 220 = 1,048,576− # of bad sequences: C(20,0) + C(20,1) = 21# of good sequences: 1,048,555

Sock Questions

You have 5 blue socks and 3 white socks in your sock drawer atrandom. 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?

Multiple Steps

Multi-Step StrategyTo do a complicated problem, try to break it up into a sequenceof smaller choices. Then we’ll use the multiplicative principle tocombine those smaller numbers.

A Short Summary

We’ve learned several principles for attacking these countingproblems.

Multi-Step Strategy: Break the problem up into asequence of smaller choices. (Once we figure out thosesmaller numbers, we can multiply them together.)Additive Principle: If you can do X or Y, count thenumber of ways of each of them and add.Change the words “at least” and “at most” into severaloptions with “or.” For example, “at most 3” means “0 or 1 or2 or 3.”Complement Principle: If counting the good ways seemshard, maybe it would be easier to count the bad ways andsubtract.

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?