1. Chapter 3. Sets & Combinatorics
Section 3.2 Counting

2. Combinatorics is a branch of mathematics that deals with counting
Combinatorics is a branch of mathematics that deals with counting. Counting is important:
e.g. How much money I have?
What’s the size of my computer memory?
How may elements (integers) are between i and j?
How many combinations to arrange 3 numbers? ……

3. e. g. A child can choose one jellybean out of
e.g. A child can choose one jellybean out of jellybeans(red, black), and one gummy bear out of three gummy bears(yellow, green, white). How many different sets of candy can the child have?
2 stages:
Y: R, Y
R G: R, G
W: R, W
choose choose
jellybean gummy bear
Y: B, Y
B G: B, G
W: B, W
6 outcomes

4. Multiplication principle:
n1 outcomes for 1st event,
n2 outcomes for 2nd event.
Then there are n1 * n2 outcomes for
the sequence of the 2 events.
e.g. The last part of your phone # contains 4 digits.
How many four-digit numbers are there?
(Select a password of 4 digits, how many
possible passwords?)
9933, 1982, 2017, 1016, 0007, …
10 * 10 * 10 * 10 = 10,000 different numbers
outcomes

5. e. g. 10 toppings available in a pizza store, and the customer
e.g. 10 toppings available in a pizza store, and the customer can choose any # of toppings, how many different pizza the store can make?
Toppings …
choose it * * * … * = 210
or w/o it =1,024
(In Section 3.4 Permutations and Combinations)

6. Addition principle:
If A and B are disjoint events with n1 and n2 possible outcomes, respectively, then the total # of outcomes for event A or B = n1 + n2.
e.g. A customer wants to buy a vehicle from a dealer. The dealer has 23 autos and 14 trucks in stock. How many selections the customer have?
= 37

7. e.g. | A  B | = |A| + |B| if A and B are disjoint sets.
In general, | A  B | = |A| + |B| - | A  B |
| A – B | = |A| - | A  B |
(In Section 3.3 Principle of Inclusion and Exclusion)
e.g. How many four-digit number begin with a 4 or 5?
2 disjoint cases:
numbers begin with 4 = 1*10*10*10=1000
numbers begin with 5 = 1000
The answer is = 2000

8. decision tree: e.g.: use x, y and z to construct strings of length 3.
How many strings do not have a z following y. x z y z x y z x y x y z x z z x y x x y z x y x z x x y y y y y
= 21