Permutations and Combinations
What are permutations? Combinations?
In what types of situations would you apply each one?
Your family is ordering an extra-large pizza. There are four toppings to choose from (pepperoni, sausage, bacon, and ham). You have a coupon for a three-topping pizza.
1.) Determine all the different three-topping pizzas you could order. You may want to create a list, diagram, table, or chart to show possible outcomes and counting techniques.
2.) Think about the pizza topping combinations you found in the previous lesson. You chose three toppings from four. Determine how many ways you can assemble a pizza with ONLY three toppings (pepperoni, sausage, bacon). This will depend on the order that ingredients are placed on the pizza. For example, putting on pepperoni, then sausage, then bacon is different than putting on bacon, then pepperoni, then sausage. Show how you determined your list.
(Adapted from www.omegamath.com)
Suppose you work at a music store and have four CDs you wish to arrange from left to right on a display shelf. The four CDs are hip-hop, country, rock, and alternative (shorthand: H, C, R, A). How many options do you have?
Solution: If you select H first then you still have three options remaining. If you then pick C, you have two CDs to choose from. You can find the number of ways to arrange your display by the factorial rule: for the first choice (event) you have 4 choices; for the second, 3; for the third, 2; and for the last, only 1. The total ways then to select the four CDs are: 4! = (4)(3)(2)(1) = 24. Factorial Rule: For n items, there are n! (pronounced n factorial) ways to arrange them. n! = (n)(n - 1)(n - 2). . . (3)(2)(1)
3! = (3)(2)(1) = 6
4! = (4)(3)(2)(1) = 24
5! = (5)(4)(3)(2)(1) = 120
6! = (6)(5)(4)(3)(2)(1) = 720
Try solving this problem:
How many ways can six different radio commercials be played during a one-hour radio program?
Permutation – A permutation is used to describe a counting procedure in which order matters.
If order matters, then AB is not the same as BA.
Consider the following:
If you have three friends to send text messages to, Taylor, Justin, and Aubrey, how many different ways can you text your friends if order matters?
Therefore TJA represents texting Taylor first, then Justin, and then Aubrey.
Because order matters, another way to text your friends is TAJ.
All of the possible arrangements are:
TJA, TAJ, JTA, JAT, ATJ, AJT
OR, 3! = (3)(2)(1) = 6 ways
Try solving this problem:
If you have four friends to send text messages to, Taylor, Justin, Arsenio, and Aubrey, how many different ways can you text your friends if order matters?
”If you want to arrange n objects in groups of n at a time, there are ___! ways to accomplish this task.
Property: There are _________ ways to arrange n objects in groups of n at a time.”
Now, let’s say you have four friends, but only need to text three of them when order matters. In order to find the number of arrangements you must use the permutation formula.
n = the total number of items you have from which to choose
r = the number you are actually going to use
Can r be larger than n? Explain your answer.
Can they be equal? Explain your answer.
Let’s say you have four friends, but only need to text three of them when order matters. Find the number of ways to text your friends.
There are 24 ways to test three out of your four friends if order matters.
Try this problem:
How many different ways can a city building inspector visit five out of six buildings in the city if she visits them in a specific order?
Combination - A combination is a counting situation in which order is not important....
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