University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
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University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
In statistics, in order to find the number of possible arrangements of a set of objects, we use a concept called permutations. There are methods for calculating permutations, and it's important to understand the difference between a set with and without repetition. Other important concepts that can apply to situations like permutations are the fundamental counting principal and basic probability.
So permutations are a way of saying different ways to arrange something okay so you basically have the same elements but by rearranging them you have a different permutation. First thing I want to do is just give you an example to visualize this. And what we're looking at is a number of ways to arrange the letters a, b and c. Okay so I'm going to start out by just writing them out, so we could have a, b, c we could just keeping our a first and then switch our b and our c have a, c, b. There's no other way we could do a first so no we can go to b first, we could say b, a, c let's switch those first 2 letters and have b, c, a and lastly could have c first and then we could have a, b or c and then b, a.
We're not talking words per say, we're just taking about the total number of arrangements so what we're left with is 6. Another way we could look at this is by just sort of making a little word, so let's say we have 3 letters and thinking about the number of letters that can go in each spot okay. We have 3 letters at our disposal so the first spot can be filled up by 3 letters, we've already used one so then the second spot could be left by 2 and the last spot there's just one letter left. Using our fundamental counting principle we can as multiply these together and end up with 6, okay obviously this way is a little bit easier but I just wanted to go through the work so you can see how this 6 actually come up okay.
This is called a factorial okay and basically what we're dealing with is the number of permutations, the number of different ways of writing the same set of objects is just n factorial. Where factorial is taking the number you're dealing with and multiplying it by every number less than that. So the number we had over here was just 3 factorial, 3 factorial is just 3 times 2 times 1 which is 6 okay. So that's what happens if we are involving every single element that we're working with.
That doesn't always take place, so over there we had 3 letters and we're arranging them into a 3 letter "word". What could also happen is we are eliminating some of those okay and for this one what we're looking at is how many different ways are there to make a 4 song playlist from 10 songs? Okay I'm going to go to the line method as opposed to writing them all out just to save a little work and so what we end up with is there are just going to be 4 slots available to us. Typically you don't include the same song twice on a playlist so we're going to deal with that duplication and so the first slot could be 1 of 10 songs. So we had 10 options there, the second spot we've already chosen one song so we're left with 9, so we're just going to throw 9 in there. The third we've chosen 2 songs already leaving us with 8 and lastly we have 7. So the number of different 4 songs play list from 10 songs is just going to be 10 times 9, times 8, times 7.
We can plug that into your calculator but I'm not entirely concerned with the actual, what the numeric value is okay. There's another way of doing this using factorials, using these little exclamation points that we talked about a second ago. And what that is, it's called the permutation formula, and how that basically works is if you have 10 objects, or n objects rather and you're permuting them choosing r of them what you end up with is n factorial over n minus r factorial. Going to this problem we'll do the same exact problem what we're looking at then is let's grab a different color 10 permute 4, okay we're choosing out of these 10 songs we're taking 4 of them which is going to work out to be 10 factorial divided by 10 minus 4 factorial or 6 factorial.
If we were to write this out what we would end up with is this is 10 times 9 times 8 so on and so forth and the bottom is 6 times 5, times 4, so on and so forth. What's going to end up happening is the 6, 5, 4, 3, 2 and 1 are going to cancel from the top and the bottom just leaving us with the 10 times 9, times 8, times 7 okay. So really we didn't even have to write out all of these little slots we could just say okay 10 permute 4 we're done. That's the general introduction to permutations. Permutation is the number of ways that you can take a set of things and rearrange them in a different way.
Unit
Topics in Discrete Math