Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Permutations and combinations. The fundamental counting principle, the number of ways in which a series of successive things can occur, is found by multiplying the number of ways in which each thing can occur. So an example of this, say you had a multiple choice quiz and you had four possible answers on five questions this. So the first question you could have four different answers. The second question, four different answers. The third question, four different answers. So you'd have 4 to the fifth different ways that you could actually answer this quiz, or 1024 different combinations that could occur, different ways to answer permutations of N things taken R at a time. The number of possible permutations if our items are taken from N items is NPR, where we have N factorial over the quantity n -, r factorial. A permutation is an ordered arrangement of items that occurs no items used more than once. The order of arrangement does make a difference. So when we look at this, oh, we're going to do some examples in a moment. The combination of N things taken R at a time is NCR, which equals N factorial over n -, r factorial times R factorial. The items are all selected from the same group. No item is used more than once and the order of the items makes no difference. So when we look at these, we're going to look at some examples. Oh, another way to think of NCR is NPR divided by R factorial or N factorial times n -, r factorial times R factorial. So if we have 7P4, we get 7 factorial over 7 - 4 factorial, 7 * 6 * 5 * 4 * 3 factorial over three factorial, or 840. If we have 9P9, we have 9 factorial over 9 - 9 factorial or 9 factorial over 0 factorial, which is 362,880. If we have 6P0, we have 6 factorial over 6 - 0 factorial or one. If we look at some combinations we have 7 choose 47 the combination 47C47 factorial over 7 - 4 factorial times 4 factorial. So 7 * 6 * 5 * 4 factorial over 3 * 2 * 1 * 4 factorial. That reduces to 35 nine C99 factorial over 9 -, 9 factorial times 9 factorial, which would give us one look at those couple examples. When we had 7P4, we ended up with 8:40, but when we had 7C4, we ended up with 35 when we had 9 P nine 362,889 choose 9/1. So we're going to look at a couple more examples. 9 bands have volunteered to perform at a benefit concert, but there is only enough time for five of the bands to play. How many lineups are possible? So there's 9 bands. This is going to be a permutation and we're going to be choosing 5 of the bands. So we have 9 factorial over 9 - 5 factorial. So 9 * 8 * 7 * 6 * 5 * 4 factorial over 4 factorial. If we compute that we're going to get 15,120. So how many lineups, IE the order is important. If band A plays first or band A plays last, we have 1015 thousand 120. If we decided that we didn't care how the lineups were, we were just wondering how many different combinations there could be for the bands, then we would have the 9 choose 5, and we'd have 9 factorial over 9 -. 5 factorial times 5 factorial. So 9 * 8 * 7 * 6 * 5 factorial over 4 * 3 * 2 * 1 * 5 factorial. 5 factorials. Cancel 4 * 2 is 8/3 goes into 6 twice, so 9 * 763 * 2 a 126. So the permutation, it mattered which band came first, which band came second, which band came third, the combination. It didn't matter which band, it just mattered that these five played or these other five played. If we look at the next example, to win at Lotto in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers one through 53. The order in which the selection is made does not matter how many different selections are possible. So we need 53 choose six, or a combination of 6. So 53 factorial over 53 -, 6 factorial times 6 factorial. So when we compute that we're going to get a bunch, we would get 53 * 52 * 51 times 50 * 49 * 48 times 47 factorial 53 - 6 is 47 factorial times 6 * 3 * 2. Oops 6 * 5 * 4 * 3 * 2 * 1. So these 47 factorials are going to cancel. 6 * 4 is 24 * 2 is 48 and the five could go into the 50, leaving us A10 and the three could go into the 51, leaving US 17. So then if we took 53 * 52 * 17 * 10 * 49 we would get to 295-7480. So there's 22,957,480 different selections that could occur. Thank you and have a wonderful day.