Subject Maths
Permutations and Combinations
Questions:
What are permutations? Combinations?
In what types of situations would you apply each one?
Launch:
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.
Investigation:
(Adapted from www.omegamath.com)
Example 1:
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)
For example:
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
Note: 0!=1
Try solving this problem:
How many ways can six
Questions:
What are permutations? Combinations?
In what types of situations would you apply each one?
Launch:
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.
Investigation:
(Adapted from www.omegamath.com)
Example 1:
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)
For example:
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
Note: 0!=1
Try solving this problem:
How many ways can six