P - Permutations and Combinations Lesson

Math_Lesson_TopBanner.png Permutations and Combinations

You probably have used the Fundamental Counting Principle before. It states that if there are n ways of doing one thing and m ways of doing another, then there are nm ways of doing both things.

Example 1

You are at a Taco Bar and there are three options for the type of shell, 4 options for the type of filling, and 6 options for toppings. That means there are LaTeX: 3\cdot4\cdot6=723⋅4⋅6=72 different types of tacos you could make!

Example 2

Let's consider another type of problem: Georgia license plates consist of 3 letters and 4 numbers. Each letter and number can repeat. How many different Georgia license plates can be made? Watch this video to determine the answer:

You Try: You have 4 paintings, in how many different ways can you hang them on the wall?

Answer: 24

Let's discuss the painting problem from above. To get 24, we multiplied, LaTeX: 4\cdot3\cdot2\cdot14⋅3⋅2⋅1.  For the first painting there were 4 options, the next 3 options, then 2, and then 1. In math, we have another way of writing LaTeX: 4\cdot3\cdot2\cdot14⋅3⋅2⋅1 and it is 4!

Factorial: multiply a series of descending natural numbers n!=n*(n-1)*(n-2)*...2*1

For example, if we have 16 pool balls, how many different ways can we order them?

We have 16 options for the first spot, let's say the 8 ball gets chosen first.

Now for the next spot, we only have 15 options because the 8 ball has already gotten picked.

So, our answer is LaTeX: 16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot116⋅15⋅14⋅13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1, which is the same as writing 16!  16! =  20,922,789,888,000. That's a lot of ways!

This is a type of permutation. A permutation represents the number of ways to order a group of items. So, there are 20,922,789,888,000 ways to order 16 pool balls. But, what if we only wanted to order 5 of the balls? Then we would only need to multiply LaTeX: 16\cdot15\cdot14\cdot13\cdot1216⋅15⋅14⋅13⋅12 so using 16! may seem like it wouldn't work, but in fact, we have a special formula for permutations that helps us out.

Permutation: The number of ways to order r items from a group of n P(n,r)=n!/(n-r)!

 

 

So, if we have 10 books, we can arrange 4 of them on a shelf in 5,040 ways. There are 10 choices for the first slot, 9 for the second, 8 for the third, and 7 for the fourth space. However, what if we just wanted to know how many groups of 4 could we take on a road trip? We wouldn't need to worry about the order the books were placed in the bag, because they are all going on the trip. If this is the case, we would not use a permutation, but a combination.

Combination: The number of ways to pick r items from a group of n C(n,r)=n!/(n-r)!r!

So, that would be LaTeX: _{10}C_4=\frac{10!}{6!\cdot4!}=21010C4=10!6!⋅4!=210

This formula looks similar to the permutation, but it divides out the 4! because that takes out the ways to arrange 4 things!

Math_PrecalculusBottomBanner.png IMAGES CREATED BY GAVS