May 08, 2015

§12.6 - Counting Principle, Permutations, and Combinations

The Fundamental Counting Principle

• Let E1 and E2 be two independent events. The first event E1 can occur in m1 ways. The second event E2 can occur in m2 ways.

• The number of ways that the combination of the two events can occur is m1m2

Ex: You are going on vacation and bringing three shirts, two pants and two pair of shoes. How many different outfits can be worn?

• Can be extended to more than two events.

• How we can find the number of ways to accomplish 2 tasks.

May 08, 2015

Try on your own:

Find the # of possible meals served at a wedding:

Appetizers: Calamari, fried mozzerella sticks, meatballs, cheeses

Soup or Salad

Entrees: Pasta, Meat or Fish

Desserts: Cake or Ice cream

There are two other ways of counting how many ways things can be ordered:

PERMUTATIONS (order matters) and COMBINATIONS (order does not matter)

Example of Permutation - if betting on a trifecta at the Kentucky Derby - to win you must not only select the first, second, and third place horses, you must select them in order in which they finished.

Example of Combination - The winning lotto numbers for one week are 1-3-5-11-18-20, the order in which the numbers are drawn doesn't matter

May 08, 2015

PERMUTATIONS

A permutation is an ordered arrangement of distinct objects without repetition.

THE NUMBER OF PERMUTATIONS OF n OBJECTS is

Ex: How many permutations of the symbols AEIOUZ are possible?

Ex: If there are 12 runners competing in trials, how many possible running orders are there?

Try on your own:

On "So you think you can dance?", eight contestants take their turns on the dance floor. How many ways can they be ordered 1-8?

The number of permutations of n objects taken r at a time is

Ex: A president, vice president, treasurer, and secretary are chosen from a pool of 11 candidates. How many different administrations exist?

May 08, 2015COMBINATIONS

an arrangement, without specific order, of distinct objects without repetition.

n objects taken r at a time

n

Ex: A virus detection program randomly samples 12 words from an email to determine if it is spam. If a typical email contains 200 words, how many different ways can the program test the email?

Ex: Towels are on sale "Buy 4 get 1 Free." There are 36 different towels. How many ways can a customer buy 5 different towels?

Try: If there are 39 possible numbers and the lottery officials draw 5 numbers, how many possibilities are there?

A permutation in which SOME of the objects ARE repeated is called

PERMUTATION WITH REPETITION or a NONDISTINGUISHABLE PERMUTATION

Drawing numbers 1-6 from a bag is different than drawing blue and red marbles from a bag. The numbers 1-6 can NOT be repeated but the colors blue and red can!

NUMBERS -

MARBLES - 6 marbles, 3 red, 2 blue, 1 white

May 08, 2015

Try: Suppose a similar game to the peg game at Cracker Barrel is set up with only ten holes in a triangle. With 4 red pegs, 2 white pegs, and 3 blue pegs, how many different permutations can fill that board?

May 08, 2015

§12.6

pg. 1174 #25-31 odds

#37-47 odds