Multiplication Rule, Permutations and Combinations
Multiplication:
· Suppose Friendly’s has a $6.99 dinner option where you get to pick a soup, a sandwich, and a milk shake.
You have 2 choices for soup: Clam Chowder and Tomato Soup
You have 3 choices for sandwiches: Grilled Cheese, Turkey, and BLT
You have 3 choices for milk shakes: Vanilla, Chocolate, and Strawberry
If you are allowed to pick one soup, one sandwich, and one milk shake,
how many different dinner combinations are there? ___________
· How many different three-digit numbers can be formed from the numbers {1,2,3,4,5,6,7,8,9}? __________
· How many different three digit numbers can be formed from these numbers if the digits in the number are to be different (i.e., 323, 233, or 333 are not allowed)? __________
· Suppose the New Jersey license plates have 3 numbers followed by 3 letters. How many different license plates are possible this way? _________
Permutations and Combinations:
A permutation of a set is an ordering (or ranking) of the members of the set.
An example would be the results when people vote for President of the United States. We write down the order in which the candidates finished. That is a permutation of the set of candidates for President.
When George W. Bush, Al Gore, and Ralph Nader ran for President, we had an ordering of 1st Bush, 2nd Gore, 3rd Nader.
We might ask how many possible orderings were there.
The answer is 6 because we have:
3 choices for who finished in first
2 choices for who finished in second (knowing that, say, Bush finished first)
1 choice for who finished in 3rd (with say Bush 1st and
Gore 2nd )
Thus our total choices for ordering is 3۰2۰1 = 6
A combination is when we take members of a set, but we DO NOT care about ordering.
An example would be for you pick which three meats among
Roast Beef (R), Ham (H), Turkey (T), Pepperoni (P), and Salami (S) to put on your sandwich. It does not really matter which order you put the meats on your sandwich.How many ways can we do this?
(RHT, RHP, RHS, RTP, RTS, RPS, HTP, HTS, HPS, TPS) 10 ways!
This is because there are:
5 ways to choose the first meat
4 ways to choose the second meat
3 ways to choose the third meat
but this gives us 5۰4۰3 = 60. What happened?
This is because order did not matter.
RHT is the same as RTH, HRT, HTR, TRH, and THR. Thus we have to divide our answer by 6.Suppose your favorite pizza place has 7 toppings:
Pepperoni, Sausage, Meatball, Bacon, Onion, Mushroom, and Broccoli
· Suppose you are to rank the 7 toppings from
your favorite to your least favorite.
How many different rankings are there?
· Now, what if you were asked to rank your
favorite 3 toppings?
How many ways can this be done?
· How many different pizzas can be made using 3 of the 7 available toppings? (Does order matter?)
Challenge questions:
· If there are 10 people at a meeting and every person shakes hand with every other person in the room except herself, how many handshakes occur?
· Suppose Mrs. Zavaglia is throwing an end of the year party where she is serving sundaes formed from two scoops of ice cream and one topping.
You can choose from the 6 ice cream flavors: Vanilla, Chocolate, Strawberry, Coffee, Pistachio and Mint Chocolate Chip. You get to choose from the 3 possible toppings: Chocolate Chips, Walnuts, and Whipped Cream.We want to find out how many different possible sundae combinations are possible. We need to know the answer to three important questions first.
1) Can a student have two scoops of the same flavor?
2) Can a student have no topping on the sundae?
3) Is a coffee/vanilla sundae different than a vanilla/coffee sundae?
a.) How many different ways can we pick the ice cream? _____
b.) How many different ways can we pick the topping? _______
c.) How many different ways can we pick the sundae? _________