Chris Morgan, MATH G160 January 30, 2012 Lecture 9 Chapter 4.1: Combinations 1

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In some problems (e.g. dealing cards) we do not care about the order that the objects are in. In this case, we deal with combinations rather than permutations: – If order matters Permutations – If order does not matter Combinations Since order doesnt matter, then we are counting some objects multiple times: Here we take into account the fact that we are counting r multiple times. 3

Does order really matter? Say we have three balls in a box – one red, one black, and one white; and we randomly choose 2 balls from the box, one at a time: If order matters, then totally we have 6 possible outcomes: {white, red}, {red, white}, {white, black}, {black, white}, {red, black}, and {black, red } If order does not matter, then totally we have 3 possible results: {white, red}, {white, black} and {black, red} 4

Calculating Combinations There should be a button on most calculators to perform permutations. There should also be a button allowing you to perform combinations. If there is not, all calculators should have a factorial button (!) which allows you to compute using the formula. 5

Combinations example (Ia) A child has 7 different toys in his toy box. He is only allowed to take three of his toys with him on a family outing. How many different sets of toys can he take? – Suppose his toys are T1, T2,T3, T4, T5, T6, T7 Remember: order does NOT matter – Therefore, choosing (T1& T2) is the same as choosing (T2& T1) 6

Combinations example (Ib) Since there are few enough possibilities, lets list them just for fun: T1, T2 T2, T3 T3, T4T4, T5 T5, T6 T6, T7 T1, T3 T2, T4 T3, T5T4, T6 T5, T7 T1, T4 T2, T5 T3, T6T4, T7 T1, T5 T2, T6 T4, T7 T1, T6 T2, T7 T1, T7 Notice, for example, that T1,T2 is counted, but T2,T1 is not counted. This is because the order does not matter; therefore, they are treated as the same event. 7

Combinations example (II) The U.S. Senate consists of 100 senators, 2 from each of the 50 states. A committee consisting of 5 senators is to be formed. - How many different committees are possible? - How many are possible if no state can have more than 1 senator on the committee? 8

Combinations example (III) What is the probability to have at least one king in a 5-card poker hand from 52 cards? - Number of poker hands? - Number of hands with at least one king? [Hint: First find the number of hands with no king and then find the compliment…] P (at least one king) 9

Ordered Partitions 10

Ordered Partitions example (17) Suppose you have six different types of flowers and three planters. In how many ways can you put three flowers in the first planer, two flowers in the second planter, and one flower in the final planter? 11

Ordered Partitions example (17b) Flowers: F1, F2, F3, F4, F5, F6 Planters: P1, P2, P3 Planter 1Planter 2Planter 3 F1, F2, F3F4, F5F6 F1, F2, F3F4, F6F5 F1, F2, F3F5, F6F4 F1, F2, F4F3, F5F6 Etc…. 12

Ordered Partitions example (17c) How many ways are there? We can refer to this as a multinomial coefficient 13

Ordered Partitions example (18a) What is the probability that in a game of Hearts, one player gets all of the Hearts? Hearts is played with 4 people; each player is dealt 13 cards from a standard 52-card deck. 14

Ordered Partitions example (18b) Total possible dealings: One player gets all the hearts: P(one player gets all hearts) = 15

Remember… Always more than one way to solve a problem If you dont know which method to use to solve a problem, thats OK! There is not always one definitive method! A good approach is to imagine yourself doing the event described When you want to count the possible outcomes, keep very close track to ALL the decisions you have to make and apply the basic counting rule General rule of thumb: - choosing with replacement BCR - choosing without replacement, order matters Permutations - choosing without replacement, order does not matter Combinations 16

More examples… A fair coin is flipped 8 times, what is the probability to see exactly 4 heads in the 8 tosses? First, find number of total possible flips: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2 8 = 256 Then, find possibilities with exactly four heads: NOT 2 * 2 * 2 * 2! Why not? P(exactly 4 heads) = 70/256 =

More examples… What is the probability to be dealt a poker hand with two Aces? First, find total number of poker hands: Then, find possibilities with exactly 2 Aces: P(2 aces) = 103,776/2,598,960 =

More examples… If five fair die are tossed, what is the probability they will all show different faces? First, find total number of outcomes: 6 * 6 * 6 * 6 * 6 = 6 5 = 7,776 Then, find possibilities (number of die) with different faces: 6 * 5 * 4 * 3 * 2 = 6! = 720 P(different faces) = 720/7,776 =

More examples… 4 tennis players (two teams of two) are to be chosen from 12 players at a club. Two of them really dislike each other and refuse to be on the same court as the other one. How many possible combinations of four players are possible? [Hint: When and why do we add and/or multiply?] We can add together the combinations of when those two players arent on the court and when they are on the court: 20

More examples… What is the probability of winning a jackpot in a lottery that requires that you have 5 correct numbers out of 30 numbers? 21

A final example… An ordinary deck of 52 cards is shuffled and dealt. What is the probability that: (a) the seventh card is an ace? (b) the first ace occurs on the seventh card dealt? 22

A final example… (a)the seventh card is an ace? First, look at all 7 card combos: [notice we used a permutation because order matters] Then, the probability the last card is any specific card: So we take: P(7 th card is an Ace) = 23

A final example… (b) the first ace occurs on the seventh card dealt? Possible outcomes where 1 st Ace is on the 7 th card? P(1 st Ace is on 7 th card) = =