Permutations & combinations
- 1. Permutations & Combinations How do you count arrangements?
- 2. When arranging objects… The fundamental counting principle gives you the number of ways a task can occur given a series of events. Suppose you have 5 students going to the movies: Adam, Brett, Candice, David and Eva. They are going to sit in 5 consecutive seats in one row at the theatre. To fill the first seat, any of the 5 students can choose to sit in that seat. After the first student is seated, any of the 4 remaining students can choose to be seated in the next seat.
- 3. Fundamental Counting Principle We continue in this manner until we get to the last seat which is left up to the one remaining student. The Fundamental Counting Principle says that we would multiply the number of ways you can fill each seat (an event) to get the total number of orderings. The solution: 5*4*3*2*1 = 120 ways you can seat the 5 students in the 5 chairs at the theatre.
- 4. Let’s count license plates! Here’s another example using FTC. Suppose you want to find the number of possible license plates for cars in North Carolina. The first three positions on the license plate are for letters and the last four positions are for digits. All 26 letters of the alphabet may be used and all ten digits, 0 – 9. Letters and digits may be repeated.
- 5. That’s a whole lotta plates! FTC tells us to multiply the number of ways you can fill each slot on the license plate. Each of the first 3 slots can be filled 26 different ways by each of the 26 letters of the alphabet. That means there are 26*26*26 = 17,576 arrangements of the 3 letters alone. Each of the next four slots can be filled by one of the ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So the ten digits have 10*10*10*10 = 10,000 different arrangements, ranging from 0000 to 9999. The total number of plates? Why just multiply together these two numbers: 175,760,000 license plates.
- 6. Factorials Before we jump into permutations & combinations, you need to understand factorials. A factorial is mathematical calculation, like square root or addition, and is represented with a ! mark. n! is the product of the numbers from n down to 1 in the decreasing sequence. It is written as n! = n*(n-1)*(n-2)*….*2*1. So 5! is 5*4*3*2*1 or 120 and 8! = 8*7*6*5*4*3*2*1. We will use factorials in permutations & combinations.
- 7. Permutations vs Combinations Permutations are written as nPx where n is the number of total choices possible and x is the number of choices that will be used. This is calculated as nPx = n! (n–x)! Permutations represent the number of ways we can choose x objects from n possibilities where the order of selection matters. Combinations represent the number of ways we can choose x objects from n possibilities where the order of selection does not matter. This is calculated as nCx = n!___ x!(n-x)!
- 8. Using the calculator for ! Your calculator has built-in functions for permutations & combinations & factorials. When you use nPx or nCx, you have to enter the n value into your calculator first, then go to Math ProbnPx to enter the function. Type in the value for x last and hit Enter. 10 P7 = 10! = 10! = 720 (10-3)! 7!
- 9. Combination example Suppose you are going out to dinner with friends. The restaurant advertises a 2-for $20 special where you may choose from one of 5 appetizers, 2 entrees from 10 possible and one dessert from 4 possibilities. You and your friend want to get different entrees. How many different ways can you choose the two entrees? Whichever of you that will choose first will have 10 selections to choose from and the other person will have 9 selections to choose from. This is calculated using permutations. 10 C2 = 10! = 10! = 45 meal combinations 2!(10-2)! 2! 8!
- 10. Permutation example Permutations are like combinations but here the order is important. Suppose you have 10 students running in a race and the top 3 winner receive medals for 1st, 2nd and 3rd places – a gold, a silver and a bronze. You calculate the number of ways this race can be won by 10 C3 = 10! = 10*9*8 = 720 ways (10-3)!
- 11. Summary Fundamental Order DOES counting principle matter. Repeats ARE allowed. Permutations Order DOES matter. Repeats are NOT allowed. Order does NOT Combinations matter. Repeats are NOT allowed.