Permutations and CombinationsArranging items and factorialsPermutations and CombinationsPermutationsCombinationsSummaryAdditional ExamplesIntroductory Statistics LecturesPermutations and CombinationsProbability IIIAnthony TanbakuchiDepartment of MathematicsPima Community CollegeRedistribution of this material is prohibitedwithout written permission of the author© 2009(Compile date: Tue May 19 14:49:10 2009)Contents1 Permutations and Com-binations 11.1 Arranging items andfactorials . . . . . . . . 11.2 Permutations andCombinations . . . . . . 2Permutations . . . . . . 2Combinations . . . . . . 31.3 Summary . . . . . . . . 41.4 Additional Examples . . 41 Permutations and Combinations1.1 Arranging items and factorialsQuestion 1. If you have 10 items, how many ways can you arrange them?(Think of a tree diagram.)factorial notation x!. Definition 1.1Denote product of decreasing whole numbers from x to 1:x! = x · (x − 1) · (x − 2) · (x − 3) · · · 3 · 2 · 1 (1)Note: 0! = 1Factorial rule. Definition 1.212 of 6 1.2 Permutations and Combinationsn items can be arranged in n! ways.Factorial:factorial(x)Finds x!(There is a limitation on how large x can be.a)aThe factorial function cannot compute values beyond x ≈ 170 due to how it’simplemented using the gamma function. The lfactorial(x) function can do largernumbers, it returns ln(x!).R CommandExample 1. To find 10! in R:R: f a c t o r i a l (1 0)[ 1 ] 36288001.2 Permutations and CombinationsPERMUTATIONSQuestion 2. How many ways can you select k = 4 students out of n = 10 whenorder matters?Anthony Tanbakuchi MAT167Permutations and Combinations 3 of 6Permutations. Definition 1.3The number of ways (permutations) that you can select k items fromn total items (all unique) when order matters is:nPk=n!(n − k)!(2)COMBINATIONSQuestion 3. How many ways can you select k = 4 students out of n = 10 whenorder does not matter? (Hint: how many ways can you arrange 4 items?)Anthony Tanbakuchi MAT1674 of 6 1.3 SummaryCombinations.Definition 1.4The number of combinations (when order does not matter) of k itemsselected from n different items without replacement is:nCk=nk=n!(n − k)!k!(3)We will need this for binomial probabilities!Combinations:choose(n, k)Computes number of combinations of k items chosen from a total ofn items.R CommandExample 2. To find10C4, the number of combinations of 4 students chosenfrom 10:R: ch oo se (1 0 , 4)[ 1 ] 2101.3 Summary• n items can be arranged in n! ways.• Number ways you can select k of n items:– Permutations: when order mattersnPk=n!(n − k)!– Combinations: when order is unimportantnCk=nk=n!(n − k)!k!nCk= choose(n,k)1.4 Additional ExamplesA Poker hand consists of 5 cards dealt from a deck of 52 cards. (A deck has2-10, J, Q, K, A — 13 difference valued cards, with 4 suits — 4 of each face.)Anthony Tanbakuchi MAT167Permutations and Combinations 5 of 6Question 4. How many simple events are there in a sample space for a pokerhand?Question 5. A royal flush is a hand that contains A, K, Q, J, 10, all in thesame suit. How many ways are there to get a royal flush?Question 6. What is the probability of being dealt a royal flush?Question 7. In how many ways can you receive four cards of the same facevalue and one card from the other 48 available cards?Question 8. What is the probability of being dealt a 4 of a kind?Question 9. A businessman in New York is preparing an itinerary for a visitto six major cities. The distance traveled, and hence the cost of the trip, willdepend on the order in which he plans his route. How many different itineraries(and trip costs) are possible?Anthony Tanbakuchi MAT1676 of 6 1.4 Additional ExamplesQuestion 10. In how many ways can you receive three cards of one face valueand another two cards of another face value? (A full house)Question 11. What is the probability of being dealt a full house?Anthony Tanbakuchi