Do Now: Make a tree diagram that shows the number of different objects that can be created. T-shirts: Sizes: S, M, L and T-shirts: Sizes: S, M, L and Type: long sleeved and short sleeved Type: long sleeved and short sleeved

Academy Algebra II/Trig 14.1: Counting, 14.2: Permutations and Combinations HW: tonight- p.978(22-24, 26),p.986(31, 34-36) HW: Monday: p (38-62 even) Quiz 14.1, 14.2: Wednesday: 11/6

Fundamental Counting Principle If one event occurs m ways & another event occurs n ways, then both events occur ways. If one event occurs m ways & another event occurs n ways, then both events occur ways. You have 3 shirts, 4 pairs of pants, and 2 pairs of shoes. How many outfits can you create? You have 3 shirts, 4 pairs of pants, and 2 pairs of shoes. How many outfits can you create?

Fundamental Counting Principle How many different license plates are possible if you have 1 letter followed by 2 digits followed by 3 letters if letters and digits can repeat? How many different license plates are possible if you have 1 letter followed by 2 digits followed by 3 letters if letters and digits can repeat? How many plates are possible if letters and digits cannot repeat? How many plates are possible if letters and digits cannot repeat?

Permutations An ordering of n objects is a permutation of the objects. (Order is important) An ordering of n objects is a permutation of the objects. (Order is important) The number of permutations of n objects is n!. The number of permutations of n objects is n!.

Permutations The number of permutations of r objects taken from a group of n distinct objects is denoted by The number of permutations of r objects taken from a group of n distinct objects is denoted by, n = total # of objects, r = how many you are taking. can also be written as., n = total # of objects, r = how many you are taking. can also be written as. We will use the calculator to get these answers – from HOME screen, go to MATH menu (2 nd 5) and select probability – nPr. We will use the calculator to get these answers – from HOME screen, go to MATH menu (2 nd 5) and select probability – nPr.

Permutations 10 people are in a race. 10 people are in a race. –How many different ways can the people finish in the race? –How many different ways can 3 people win 1 st, 2 nd, and 3 rd place?

Permutations p.981 ex 3: In how many ways can 5 people be lined up? p.981 ex 3: In how many ways can 5 people be lined up?

Permutations P.982 ex 5: All we know about Shannon, Patrick, and Ryan is that they have different birthdays. If we listed all the possible ways this could occur, how many would there be? (Assume there are 365 days in a year.) P.982 ex 5: All we know about Shannon, Patrick, and Ryan is that they have different birthdays. If we listed all the possible ways this could occur, how many would there be? (Assume there are 365 days in a year.)

Permutations with Repetition The number of permutations of n objects where an object repeats s # of times. The number of permutations of n objects where an object repeats s # of times.

Find the number of distinguishable permutations of the letters in the word. 1.) WYNES 2.) TALLAHASSEE 3.) MATAWAN

Combinations An ordering of r objects from a total of n objects where order is not important is a combination. An ordering of r objects from a total of n objects where order is not important is a combination.

Combinations The number of combinations of r objects taken from a group of n distinct objects is denoted by The number of combinations of r objects taken from a group of n distinct objects is denoted by, n = total # of objects, r = how many you are taking., n = total # of objects, r = how many you are taking. We will use the calculator to get these answers – from HOME screen, go to MATH menu (2 nd 5) and select probability – nCr. We will use the calculator to get these answers – from HOME screen, go to MATH menu (2 nd 5) and select probability – nCr.

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

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

A club has a president and vice-president position. Out of 12 students, how many ways can students be chosen for these two positions? A club has a president and vice-president position. Out of 12 students, how many ways can students be chosen for these two positions?

Combination or Permutation A relay race has a team of 4 runners who run different parts of the race. There are 20 students on your track squad. In how many ways can the coach select students to compete on the relay team? A relay race has a team of 4 runners who run different parts of the race. There are 20 students on your track squad. In how many ways can the coach select students to compete on the relay team?

Combination or Permutation P.987 #53: An urn contains 7 white balls and 3 red balls. Three balls are selected. In how many ways can the 3 balls be drawn from the total of 10 balls: P.987 #53: An urn contains 7 white balls and 3 red balls. Three balls are selected. In how many ways can the 3 balls be drawn from the total of 10 balls: –If 2 balls are white and 1 is red? –If all 3 balls are white? –If at least 2 balls are red?**

Combination or Permutation P.987 #59: A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed? P.987 #59: A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed?

From a standard 52-card deck, find the number of 5-card hands that contain the cards specified. 1.) 5 of any card 2.) 5 face cards

From a standard 52-card deck, find the number of 5-card hands that contain the cards specified. 3.) 5 cards of the same color 4.) 1 ace and 4 cards that are not aces

From a standard 52-card deck, find the number of 5-card hands that contain the cards specified. 5.) 5 clubs or 5 spades 6.) at most 1 queen