MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there?

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime.

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails.

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails. HHH HHT HTH HTT THH THT TTH TTT

MATH 110 Sec 12-1 Lecture: Intro to Counting Introduction to Counting: Just how many are there? Suppose I flip 3 coins, a penny, a nickel and a dime The structure that we used to count the number of outcomes is called a TREE DIAGRAM. …beginning with the penny …then the nickel …finally, the dime There are 8 different arrangements of Heads and Tails. HHH HHT HTH HTT THH THT TTH TTT

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting FT There are 2 ways (T or F) to answer Question 1

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting FT If Q1 is answered T, Q2 can be answered either T or F. There are 2 ways (T or F) to answer Question 1

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT If Q1 is answered T, Q2 can be answered either T or F There are 2 ways (T or F) to answer Question 1

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT If Q1 is answered T, Q2 can be answered either T or F There are 2 ways (T or F) to answer Question 1

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F If Q1 is answered T, Q2 can be answered either T or F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F TF If Q1 is answered T, Q2 can be answered either T or F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT There are 2 ways (T or F) to answer Question 1 If Q1 is answered F, Q2 can be answered either T or F If Q1 is answered T, Q2 can be answered either T or F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Similarly, each possible answer to Q2 leads to 2 possible answers for Q3. Q3

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF And each possible answer to Q3 leads to 2 possible answers for Q4. Q4

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T

If you are taking a 4 question T/F quiz: MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. How many different ways are there to answer all 4 questions?

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF TF TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT

F If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TFTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF, FFFT F F

If you are taking a 4 question T/F quiz: How many different ways are there to answer all 4 questions? MATH 110 Sec 12-1 Lecture: Intro to Counting TTF FT Q1 Q2 Q3 TF T TF TF Q4 F T F T F T F T F T F T F T F T There are 16 different ways to answer all 4 questions. You can list all 16 ways by going down each of the 16 branches of the tree. { TTTT, TTTF, TTFT, TTFF, TFTT, TFTF, TFFT, TFFF, FTTT, FTTF, FTFT, FTFF, FFTT, FFTF, FFFT, FFFF} F F

MATH 110 Sec 12-1 Lecture: Intro to Counting

Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the 5 could still be chosen second. A BCDE 5 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways …and because AB and BA are considered to be different.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If B is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed A BCDE 5 ways …and because AB and BA are considered to be different. A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways Similarly, if C is the first letter chosen, any of the 5 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE 5 ways Similarly, if C is the first letter chosen, any of the 5 could still be chosen second. C A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways The same logic holds for D and E being selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE 5 ways The same logic holds for D and E being selected first. A BCDE A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE A BCDE A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE A BCDE A BCDE A BCDE A BCDE So, there are 25 ways that this can be done. 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting

Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the other 4 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A can’t be used again because repetition is not allowed. If A is the first letter chosen, any of the other 4 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE BCDE If A is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowedA can’t be used again because repetition is not allowed.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE BCDE 4 ways If A is the first letter chosen, any of the other 4 could still be chosen second. A can’t be used again because repetition is not allowed.

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE BCDE 4 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the other 4 could still be chosen second. A BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE This is because repetition is not allowed BCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE …and because AB and BA are considered to be different. BCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowed

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE …and because AB and BA are considered to be different. ACDEBCDE 4 ways If B is the first letter chosen, any of the other 4 could still be chosen second. This is because repetition is not allowed

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Similarly, if C is the first letter chosen, any of the other 4 could still be chosen second. ACDEBCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDEC ACDEBCDE 4 ways ABDE Similarly, if C is the first letter chosen, any of the other 4 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways ABDE

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. ACDEBCDE 4 ways ABDE

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. ACDEBCDE 4 ways ABDE ABCE ABCD

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE ACDEBCDE 4 ways ABDE ABCE ABCD

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE So, there are 20 ways that this can be done. ACDEBCDE 4 ways ABDE ABCE ABCD

MATH 110 Sec 12-1 Lecture: Intro to Counting

Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Any of the 5 letters could be selected first.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE If A is the first letter chosen, any of the 5 could still be chosen second. This is because repetition is allowed. A BCDE 5 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE A BCDE 5 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting BCDE If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second. A BCDE 5 ways A

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE This is because although repetition is allowed A BCDE 5 ways If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways …AB and BA are NOT considered to be different. This is because although repetition is allowed If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways BCDE This is because although repetition is allowed …AB and BA are NOT considered to be different. 4 ways If B is the first letter chosen, any of the 4 remaining letters (any but A) could still be chosen second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE Similarly, if C is the first letter chosen, any of the 3 remaining letters (any but A or B) could still be chosen. second. A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDEC CDE 3 ways A BCDE 5 ways BCDE 4 ways Similarly, if C is the first letter chosen, any of the 3 remaining letters (any but A or B) could still be chosen. second.

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE CDE 3 ways A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. CDE 3 ways A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE The same logic holds for D and E being selected first. DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting A BCDE DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting So, there are 15 ways that this can be done. A BCDE DEE 2 ways1 way CDE 3 ways A BCDE 5 ways BCDE 4 ways

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible?

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else.

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else. There are 6 x 6 = 36 entries in this table.

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. How many different rolls are possible? Although a tree diagram will work, let's try something else. Each entry in this table represents a different roll so there are 36 different possible rolls. There are 6 x 6 = 36 entries in this table.

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9?

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9?

MATH 110 Sec 12-1 Lecture: Intro to Counting Suppose you roll 2 dice. The first is red & the second is green. In how many ways can you roll a sum of 9? So there are 4 ways to roll a sum of 9.