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Page 1: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence
Page 2: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 3: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 4: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 5: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 6: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 7: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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.

Page 8: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

“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.

Page 9: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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.

Page 10: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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.

Page 11: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

An important table

or +and ×

Page 12: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 13: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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)

Page 14: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 15: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 16: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 17: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 18: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 19: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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

Page 20: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?

Page 21: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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.

Page 22: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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.

Page 23: Coin Flip Questions - Concordia Collegefaculty.cord.edu/ahendric/203/FurtherCountingProblems.pdf · Coin Flip Questions Suppose you flip a coin five times and write down the sequence

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?