Combinations

Group the following into 2 groups of similar type questions How many ways can the letters in the word “ALLIGATOR” be arranged? How many ways can you select a 5-card poker hand? How many ways can a President, Vice President and Treasurer be selected from 12 board members? How many ways can groups of 4 students be selected from the class? How many ways can you select 3 pieces of candy from a bag of 15 different kinds? How many ways could the numbers 1-7 be used to create a 5-digit number? How many possible numbers can be selected in Lotto 6/49, in which 6 different numbers must be selected? A picture is being taken of the Senior Basketball team, how many ways can the front row of 6 players be chosen from a possible 15?

Permutations v Combinations Order of objects DOES matter A – Obviously different orders of letters creates different words C – If a group of people have different roles, order does matter F – Any different arrangement of numbers also creates a unique number H – If a picture is being taken, it is assumed that any difference in appearance is to be considered Order of objects DOES NOT matter B – The order that a hand is dealt to you does not matter, only that those cards are all in your hand D – As long as there are no specific roles in these groups it is arbitrary in which order their names appear E – If you are selecting 3 candies the order that they appeared in your hand is also irrelevant G – The rules for Lotto 6/49 force you to pick 6 unique numbers from 1-49, order is not relevant

Combinations Explained 12 friends gather to play a game of Grounders, how many ways: Can the 12 players be chosen if order mattered? 12! Can the 12 players be chosen to play? Every order should be considered identical, because all 12 players are chosen in each arrangement

Combinations Explained How many ways could we select 3 kids to be “it”? __12__ x __11___ x __10___ = 1320 Does it matter which order we arranged them in? NO! Players are only concerned with whether they are “it” How many ways will we have counted each specific group of 3? __3__ x __2___ x __1___ = 6 Ex: We would have counted Billy, Sam, Jamie as Billy, Jamie, Sam and Jamie, Billy, Sam, etc.

Combinations Explained Think about our treatment of identical objects, or circular questions, how did we deal with over-counting arrangements for all possibilities? Divide by # of times each identical arrangement (to us) is counted

Over-Counting Examples SASKATOON # of arrangements # 𝑜𝑓 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑎𝑝𝑝𝑒𝑎𝑟𝑎𝑛𝑐𝑒𝑠 = 𝟗! 𝟐!𝟐! # of arrangements # 𝑜𝑓 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑎𝑝𝑝𝑒𝑎𝑟𝑎𝑛𝑐𝑒𝑠 = 𝟒! 𝟒

Combinations Explained Think about our treatment of identical objects, or circular questions, how did we deal with over-counting arrangements for all possibilities? Divide by # of times each identical (over-counted) arrangement (to us) is counted

Combinations Formula nCr = 𝑛 𝑟 = 𝑛! (𝑛−𝑟)!𝑟! If a scenario calls for us to “choose” rather than “arrange” a number of objects, we can usually assume that order does NOT matter. Notice also that we are only considering cases where we choose SOME not ALL from a group. n “choose” r: nCr = 𝑛 𝑟 = 𝑛! (𝑛−𝑟)!𝑟! Typically we will use the 2nd format to organize our solution, and the 1st to calculate.

Example A co-ed ultimate Frisbee team consists of 9 men and 7 women. How many ways can they choose the starting 7 (no unique positions) if: No restrictions apply? It must be all men? Sandra must be on the field? There must be 4 men, and 3 women?

Example Continued Sandra must be on the field? We need to think back to our FCP that tells us to multiply the # of ways to complete action 1 with the # of ways to complete action 2. - Becomes Sandra must be on the field AND 6 more players There must be 4 men, and 3 women? Again because order does not matter we can treat these actions separately and then multiply

Questions 2. Evaluate: a. 17C5 b. 17C12   a. 17C5 b. 17C12 c. Explain the similarity between these two combinations. 3. A committee of 3 is selected from 5 girls and 7 boys. How many if:   a. no restrictions? b. all boys? c. 1 girl and 2 boys? d. 1 girl or 2 boys? Careful! e. at least 1 girl? 1. Combination or Permutation? a. letter arrangements b. committees (need 5 people) c. card hands d. combination lock e. people lined up for a photo f. diagonals in a polygon g. an executive (president, VP, secretary) h. sum of postage from different stamps i. numbers (43 vs 34) 4. How many 5-card hands in cards:   a. no restrictions?   b. all hearts? c. 1 red king & 4 black cards? d.at least 1 black?