Organized Counting

Combinatorics

- is a branch of math dealing with ideas and methods for counting, especially in complex situations.

When making a series of choices, you can determine the total number of possibilities without actually counting each one individually.

Example.

Mark travels from Hamilton to Calgary and notices that he has three choices, Train, Bus, Plane. From Calgary, he wants to get to Dallas via a plane and has 3 choices. To get back to Hamilton he has two choices. How many ways can he go from Hamilton to Hamilton in a complete route?

First portion of trip 3

Second portion 3

Third portion 2

Total number 3 x 3 x 2 = 18

We can draw a diagram to illustrate this

Bus

Train

Plane

Flight B

Flight A

Flight C

Flight B

Flight A

Flight C

Flight B

Flight A

Flight C

Flight 1Flight 2

Flight 1

Flight 1

Flight 1

Flight 1

Flight 1

Flight 1

Flight 1

Flight 1

Flight 2

Flight 2

Flight 2

Flight 2

Flight 2

Flight 2

Flight 2

Flight 2

18 Possible Choices

A fast food restaurant has a menu that you get to choose your food. You get five choices of drinks, four sizes of fries and 7 different burgers. How many choices are there?

5x4x7 = 140

This illustrates counting problems called fundamental or multiplicative counting principles

Total number of choices m x n x p x o x q for m choices, n choices, o choices etc…

Recap:

Indirect MethodJen has a dresser with four pairs of different coloured socks. In how many ways can she pull out two unmatched socks one after the other

Find the answer by subtracting the number of ways of picking matched ones from the number of ways of picking two socks

There are eight possibilities when Jen pulls out the first sock, but only seven when she pulls out the second sock. By the counting principle, the number of ways Jen can pick any two socks out of the bag is 8 x 7 = 56.

She could pick each of the unmatched pairs in two ways: left sock then right sock or right sock then left sock. Therefore, there are 4 x 2 = 8 ways of picking a matched pair.

Therefore the total # of ways Jen can pull out two unmatched socks one after the other is 56 – 8 = 48 ways.

Sailing ships use sails to send signals. How many possible signals are there with a set of four distinct flags if a minimum of two flags is used for each signal?

4 choices x 3 choices = 12 Signal with two flags

4 choices x 3 choices x 2 choices = 24 Signal with three flags

4 x 3 x 2 x 1 = 24 Signal with four flags

 

Total number of signals = 60 possible signals

If you choose from m items of one type and n items of another, there are n x m ways to choose one item of each type (multiplicative counting principle)

M relates to n. You have to do m before n.

If you choose from either m items of one type or n items of another type, then the total number of ways you can choose an item is n + m (additive counting principle)

Both of these methods apply to three or more types of items.

M doesn’t relate to n in this case.

If you are counting actions that don’t occur at the same time (mutually exclusive) you can add (additive counting principle)

Homework

Pg 229 # 1, 2, 4, 5a, 9,14,16a,c,18a,b, 24a,