Combinations A combination is a grouping of things ORDER DOES NOT MATTER
How many arrangements of the letters a, b, c and d can we make using 3 letters at a time if order does not matter? We know there are 4! = 24 permutations. Listed out they are: abc acb bac bca cab cba = 1 combination abd adb bad bda dab dba = 1 combination acd adc cad cda dac dca = 1 combination bcd bdc cbd cdb dbc dcb = 1 combination = 4 combinations, total
Each of the above combinations had 3 letters, so there were 3! ways to change the order around (3! permutations). If order doesn’t matter we will divide by that number. The number of ways of choosing r objects from a set of n without regard to order is: This is commonly read as “n choose r”
Two examples to show the difference between permutations and combinations: How many seating arrangements of 6 students can be made from a class of 30? (order matters – a permutation)
How many ways are there of choosing 6 students for a class project in a class of 30? (order does not matter – just that 6 students are picked – a combination)
How many different 6 number lottery tickets can be issued? A purchaser picks 6 numbers from 00 – 99 and it does not matter which order they are in.
Picking 6 correct lottery numbers . . . 100 numbers to pick from Want the 6 that are correct
How many different 5-card poker hands are there?
Different 5-card poker hands . . . 52 cards to pick from Want 5 cards total
How many different 5-card hands can there be if all cards must be clubs (a flush in clubs)?
A flush in clubs . . . 13 clubs to pick from Want 5 cards total
How many different 5-card hands can there be if all cards must be the same suit (a flush in any suit)?
A flush in ANY suit . . . 4 cases that are the same - the 4 cases are from the 4 suits: hearts, spades, clubs or diamonds. Same as the previous example
How many different 5-card hands can there be that contain exactly 3 Aces?
5 cards with exactly 3 aces . . . # ways to get other 2 cards # ways to get 3 Aces *
How many different 5-card hands can there be that contain at least 3 Aces?
5 cards with at least 3 aces . . . # ways to get 3 Aces (from previous example) # ways to get 4 Aces + = 4512 + 48 = 4560 ways
How many different 5-card hands can there be that contain exactly 2 Hearts?
5 cards with exactly 2 hearts . . . # ways to get 2 Hearts # ways to get other 3 cards *