Aim: Combinations Course: Math Lit. Do Now: Aim: How do we determine the number of outcomes when order is not an issue? Ann, Barbara, Carol, and Dave are the only members of a school club. In how many different ways can they elect a president and treasurer for the club? Ann, Barbara, Carol, and Dave are the only members of a school club. In how many ways can they choose 2 people to represent the club at student council meetings? Explain how these situations are different.

Aim: Combinations Course: Math Lit. Subsets & Arrangements If order were important {a, b} is = {b, a} ? No A = {a, b, c, d, e} If the two elements a and b are selected from A, then there is one subset (order not important): {a, b} there are two arrangements (order important): {a, b} and {b, a} order is important A = {Ann, Barbara, Carol, Dave} AnnBarbara presidenttreasurer CarolAnnDaveCarolBarbara DaveAnn DaveCarolDaveBarbaraCarol member of council DaveCarol

Aim: Combinations Course: Math Lit. Permutation Ann, Barbara, Carol, and Dave are the only members of a school club. In how many different ways can they elect a president and treasurer for the club? Ann Barbara Carol Dave President Barbara Carol Dave Ann Carol Dave Barbara Ann Dave Barbara Carol Ann Treasurer. Ann & Barbara Ann & Carol Ann & Dave Barbara & Ann Barbara & Carol Barbara & Dave Carol & Barbara Carol & Ann Carol & Dave Dave & Barbara Dave & Carol Dave & Ann 4 P 2 = 4 3 = 12 There are 12 different arrangements of two people for president and treasurer.

Aim: Combinations Course: Math Lit. Combination Ann, Barbara, Carol, and Dave are the only members of a school club. In how many ways can they choose 2 people to represent the club at student council meetings? Ann Barbara Carol Dave 1st Person Barbara Carol Dave Ann Carol Dave Barbara Ann Dave Barbara Carol Ann 2nd Person Ann & Barbara Ann & Carol Ann & Dave Barbara & Ann Barbara & Carol Barbara & Dave Carol & Barbara Carol & Ann Carol & Dave Dave & Barbara Dave & Carol Dave & Ann There are six combinations of two people that can represent

Aim: Combinations Course: Math Lit. Order: Permutation vs. Combination A selection of objects in which their order is not important. When selecting some of the objects in the set: The number of combinations of n objects r at a time When selecting all objects in the set: there is only 1 combination!! = 1

Aim: Combinations Course: Math Lit. Combinations 1. For any counting number n, n C n = 1 3 C 3 = 1 10 C 10 = 1 2. For any counting number n, n C 0 = 1 5 C 0 = 1 34 C 0 = 1 Some Special Relationships 3. For whole numbers n and r, where r < n, n C r = n C n - r 7 C 3 = 7 C = 7 C 4 23 C 16 = 23 C = 23 C 7

Aim: Combinations Course: Math Lit. Combinations & Pascal’s Triangle 0 C 0 = 1 1 C 0 =1 C 1 = 11 2 C 0 =2 C 1 = 2 C 2 = C 0 =3 C 1 =3 C 2 =3 C 3 = C 0 =4 C 1 =4 C 2 =4 C 3 =4 C 4 = C 0 =5 C 1 =5 C 2 =5 C 3 =5 C 4 =5 C 5 =

Aim: Combinations Course: Math Lit. Combination Ann, Barbara, Carol, and Dave are the only members of a school club. In how many ways can they choose 2 people to represent the club at student council meetings? Ann Barbara Carol Dave 1st Person Barbara Carol Dave Ann Carol Dave Barbara Ann Dave Barbara Carol Ann 2nd Person Ann & Dave Barbara & Ann Barbara & Carol Barbara & Dave Carol & Barbara Carol & Ann Carol & Dave Dave & Barbara Dave & Carol Dave & Ann 4C24C2 = 4 P 2 / 2! = 6 Ann & Barbara Ann & Carol

Aim: Combinations Course: Math Lit. Model Problems Evaluate: 10 C 38C28C2 How many different three-person committees can be formed from a group of eight people? Is order important? NO 8C38C3 = 56 = 120 = 28 A committee has 7 men and 5 women. A subcommittee of 8 is to be formed. Write an expression for the number of ways the choice can be made. 12 C 8 = 495 In general, use permutations where order is important, and combinations where order is not important.

Aim: Combinations Course: Math Lit. Model Problem Is the order of the 4 marbles important? NO! Combination 457 = 2380 From an urn containing 4 black marbles, 8 blue marbles, and 5 red marbles, in how many ways can a set of 4 marbles be selected? 17 C 4 = 17 total marbles

Aim: Combinations Course: Math Lit. Model Problem If n C 2 = 15, what is the value of n? nC2nC2 n(n - 1) = 215 6C26C2 = 15 n 2 - n = 30 n 2 - n - 30 = 0 (n - 6)(n + 5) = 0 (n - 6) = 0 (n + 5) = 0 n = 6 n = -5

Aim: Combinations Course: Math Lit. Fundamental and Combinations A committee of five is chosen from five mathematicians and six economists. How many different committees are possible if the committee must include two mathematicians and three economists? mathematicians: economists: 5C25C2 6C36C3. = 10 · 20 = 200

Aim: Combinations Course: Math Lit. Model Problem The US Senate of the 104 th Congress consisted of 54 Republicans and 46 Democrats. How many committees can be formed if each committee must have 3 Republicans and 2 Democrats? Republicans: Democrats: 54 C 3 46 C 2. = 24,804 · 1035 = 25,672,140

Aim: Combinations Course: Math Lit. Model Problems There are 10 boys and 20 girls in a class. Find the number of ways a team of 3 students can be selected to work on a project if the team consists of: 30 C 3 10 C 120 C 2 20 C C 120 C 2 = 4060 = = 1910 = C 0 = = 3040 A. Any 3 students B. 1 boy and 2 girls C. 3 girls D. At least 2 girls 2 girls 10 C 0 3 girls

Aim: Combinations Course: Math Lit. Model Problem In how many ways can 6 marbles be distributed in 3 boxes so that 3 marbles are in the first box, 2 in the second, and 1 in the third Box 2Box 1Box 3 6C36C33C23C21C11C = 60

Aim: Combinations Course: Math Lit. Model Problem Find the number of ways to select 5-card hands from a standard deck so that each hand contains at most 2 aces. at most 2 aces Means that the hand could have 0, 1 or 2 aces W/ no Aces 4C04C048 C 5 W/ 1 Aces 4C14C148 C 4 W/ 2 Aces 4C24C248 C 3 Choose AcesComplete the 5-card hand = = = =

Aim: Combinations Course: Math Lit. Do Now: Aim: How do we determine the number of outcomes when order is not an issue? In the “Pick Four” Lottery, you create a 4-digit number using the numbers 1, 2, 3, 4, 5, and 6. If you play the game “straight”, you win if the winning lottery number matches your selection exactly. How many different arrangements are possible if you bet the game “straight”? 6P46P4 = 360

Aim: Combinations Course: Math Lit. Model Problems In the “Pick Four” Lottery, you create a 4-digit number using the numbers 1, 2, 3, 4, 5, and 6. If you play the game “straight”, you win if the winning lottery number matches your selection exactly. How many different arrangements are possible if you bet the game “straight”? 6P46P4 = 360 If you choose you may, you may play the game “boxed”. This means that as long as the same four numbers are chosen, regardless of order, you win. How many possible combinations are possible? 6C46C4 = 15

Aim: Combinations Course: Math Lit. Model Problem How many different 4-member committees can be formed from a group of 10 people if Tony, 1 of the 10 must: A. Always on the committee AFTER TONY IS PLACED ON THE COMMITTEE, THERE ARE 3 PLACES LEFT FOR THE OTHER 9 PEOPLE 1 = 84 9C39C3 9 C 3 = 84 TONY IS A MUST!

Aim: Combinations Course: Math Lit. Model Problem How many different 4-member committees can be formed from a group of 10 people if Tony, 1 of the 10 must: B. Never be on the committee 9C49C4 There are now only 9 possible members for the 4-member committee = 126

Aim: Combinations Course: Math Lit. Counting Techniques Tree DiagramFundamental Counting Principle CombinationsPermutations Use this to handle inconsistencies most tedious, use when all else fails Counts total number of separate tasks Repetitions not allowed Repetitions allowed Subsets Arrangements total number of ways a task can be performed m · n · o · p ··· Order does not matter Order matters

Aim: Combinations Course: Math Lit. Model Problems Sets of 2 letters are chosen from the English alphabet. Find the number of 2-letter sets possible if the set: a.cannot have a vowel b.cannot have a consonant c.must have at most 1 vowel d.must have a vowel and a consonant

Aim: Combinations Course: Math Lit. Model Problems Find the number of ways a coach can select her starting basketball team from a group of 12 players, 8 boys and 4 girls, if the positions to be played are not taken into account, and if: a.Sally, 1 of the players is always on the team b.Ed, 1 of the players is never on the team. c.both Sally and Ed are not on the team d.either Sally or Ed, but not both, is on the team.

Aim: Combinations Course: Math Lit. Model Problems Sets of 4 letters are chosen from the English alphabet. Find the number of 4-letter lets possible if there must be the same number of vowels and consonants, and if: a.A is always included b.M is always included c.E is never included d.Q is never included