Do Now: Make a tree diagram that shows Do Now: Make a tree diagram that shows the number of different objects that can be the number of different objects that can be

created.created. T-shirts: Sizes: S, M, L and T-shirts: Sizes: S, M, L and

Type: long sleeved and short sleevedType: long sleeved and short sleeved

Academy Algebra Academy Algebra II/TrigII/Trig

14.1: Counting,14.1: Counting,14.2: Permutations and Combinations14.2: Permutations and CombinationsHW: tonight-p.978(22-24, 26),p.986(31, 34-

36)HW: Monday: p.986-987 (38-62 even)

Quiz 14.1, 14.2: Wednesday: 11/6

Fundamental Counting Fundamental Counting PrinciplePrinciple

If one event occurs If one event occurs mm ways & another ways & another event occurs event occurs nn ways, then both events ways, then both events occur ways.occur ways.

You have 3 shirts, 4 pairs of pants, and You have 3 shirts, 4 pairs of pants, and 2 pairs of shoes. How many outfits can 2 pairs of shoes. How many outfits can you create?you create?

Fundamental Counting Fundamental Counting PrinciplePrinciple

How many different license plates are How many different license plates are possible if you have 1 letter followed possible if you have 1 letter followed by 2 digits followed by 3 letters if by 2 digits followed by 3 letters if letters and digits can repeat? letters and digits can repeat?

How many plates are possible if letters How many plates are possible if letters and digits cannot repeat?and digits cannot repeat?

PermutationsPermutations

An ordering of n objects is a An ordering of n objects is a permutation of the objects. (Order permutation of the objects. (Order is important)is important)

The number of permutations of n The number of permutations of n objects is n!.objects is n!.

PermutationsPermutations The number of permutations of The number of permutations of rr objects taken objects taken

from a group of from a group of nn distinct objects is denoted by distinct objects is denoted by

, n = total # of objects, r = how many you , n = total # of objects, r = how many you are taking. can also be written as . are taking. can also be written as .

We will use the calculator to get these We will use the calculator to get these answers – from HOME screen, go to MATH answers – from HOME screen, go to MATH menu (2menu (2ndnd 5) and select probability – nPr. 5) and select probability – nPr.

PermutationsPermutations 10 people are in a race. 10 people are in a race.

– How many different ways can the How many different ways can the people finish in the race?people finish in the race?

– How many different ways can 3 How many different ways can 3 people win 1people win 1stst, 2, 2ndnd, and 3, and 3rdrd place? place?

PermutationsPermutations p.981 ex 3: In how many ways can p.981 ex 3: In how many ways can

5 people be lined up? 5 people be lined up?

PermutationsPermutations P.982 ex 5: All we know about P.982 ex 5: All we know about

Shannon, Patrick, and Ryan is that Shannon, Patrick, and Ryan is that they have different birthdays. If we they have different birthdays. If we listed all the possible ways this listed all the possible ways this could occur, how many would there could occur, how many would there be? (Assume there are 365 days in be? (Assume there are 365 days in a year.)a year.)

Permutations with Permutations with RepetitionRepetition

The number of permutations of The number of permutations of nn objects where an object repeats objects where an object repeats ss # # of times.of times.

Find the number of distinguishable Find the number of distinguishable permutations of the letters in the permutations of the letters in the

word.word.1.) WYNES1.) WYNES

2.) TALLAHASSEE2.) TALLAHASSEE

3.) MATAWAN3.) MATAWAN

CombinationsCombinations

An ordering of An ordering of rr objects from a objects from a total of total of nn objects where order is objects where order is not important is a combination. not important is a combination.

CombinationsCombinations The number of combinations of The number of combinations of rr objects taken objects taken

from a group of from a group of nn distinct objects is denoted by distinct objects is denoted by

, n = total # of objects, r = how many you , n = total # of objects, r = how many you are taking.are taking.

We will use the calculator to get these We will use the calculator to get these answers – from HOME screen, go to MATH answers – from HOME screen, go to MATH menu (2menu (2ndnd 5) and select probability – nCr. 5) and select probability – nCr.

P.984 ex 8: How many different P.984 ex 8: How many different committees of 3 people can be committees of 3 people can be formed from a pool of 7 people?formed from a pool of 7 people?

Combination or PermutationCombination or Permutation

P.984 ex 9: In how many ways can a P.984 ex 9: In how many ways can a committee of 2 faculty members and committee of 2 faculty members and 3 students be formed if 6 faculty 3 students be formed if 6 faculty members and 10 students are members and 10 students are eligible?eligible?

Combination or PermutationCombination or Permutation

Combination or PermutationCombination or Permutation A club has a president and vice-A club has a president and vice-

president position. Out of 12 president position. Out of 12 students, how many ways can students, how many ways can students be chosen for these two students be chosen for these two positions?positions?

Combination or PermutationCombination or Permutation A relay race has a team of 4 runners A relay race has a team of 4 runners

who run different parts of the race. who run different parts of the race. There are 20 students on your track There are 20 students on your track squad. In how many ways can the squad. In how many ways can the coach select students to compete on coach select students to compete on the relay team?the relay team?

Combination or PermutationCombination or Permutation P.987 #53: An urn contains 7 white P.987 #53: An urn contains 7 white

balls and 3 red balls. Three balls are balls and 3 red balls. Three balls are selected. In how many ways can the 3 selected. In how many ways can the 3 balls be drawn from the total of 10 balls be drawn from the total of 10 balls:balls:– If 2 balls are white and 1 is red?If 2 balls are white and 1 is red?

– If all 3 balls are white?If all 3 balls are white?

– If at least 2 balls are red?**If at least 2 balls are red?**

Combination or PermutationCombination or Permutation P.987 #59: A baseball team has 15 P.987 #59: A baseball team has 15

members. Four of the players are members. Four of the players are pitchers, and the remaining 11 pitchers, and the remaining 11 members can play any position. How members can play any position. How many different teams of 9 players many different teams of 9 players can be formed?can be formed?

From a standard 52-card deck, find From a standard 52-card deck, find the number of 5-card hands that the number of 5-card hands that

contain the cards specified.contain the cards specified.

1.) 5 of any card1.) 5 of any card

2.) 5 face cards2.) 5 face cards

From a standard 52-card deck, find From a standard 52-card deck, find the number of 5-card hands that the number of 5-card hands that

contain the cards specified.contain the cards specified.

3.) 5 cards of the same color 3.) 5 cards of the same color

4.) 1 ace and 4 cards that are not 4.) 1 ace and 4 cards that are not acesaces

From a standard 52-card deck, find From a standard 52-card deck, find the number of 5-card hands that the number of 5-card hands that

contain the cards specified.contain the cards specified.

5.) 5 clubs or 5 spades 5.) 5 clubs or 5 spades

6.) at most 1 queen6.) at most 1 queen