15.2 – The Multiplication, Addition, and Complement Principles

1. 15.2 – The Multiplication, Addition, and ComplementPrinciples Objective: You should be able to use these principles to solve counting problems.

2. The Multiplication Principle • If an action can be performed in n1 ways, and for each of these ways another action can be performed in n2 ways, then the two actions can be performed together in n1n2 ways.

3. Example: • a. If a girl has 5 different pairs of pants and 9 different t-shirts, how many pant-and-t-shirt outfits are possible? • b. If she also has 4 hoodies, how many pant-t-shirt-and-hoodie outfits are possible?

4. Example: • In how many different orders can you arrange how you stack 6 different color blocks?

5. !!!!!!!!!!!!!!!! Factorial !!!!!!!!!!!!!!!!! n! = n(n – 1) (n – 2) … 3 2 1 9! = 9 8 7 6 5 4 3 2 1 = 362880 0! = 1

6. The Addition Principle • If two actions are mutually exclusive, and the first can be done in n1 ways and the second in n2 ways, then one action or the other can be done in n1 + n2ways. • Mutually exclusive – they cannot be performed together. (# or letter)

7. Example: • In some states license plates consist of 3 letters followed by 2 or 3 digits. If this is the case, how many license plates are possible?

8. Example: • Given the digits 0, 1, 2, 3, 4, and 5, how many four-digit numbers can be formed if: • a. digits can be repeated?

9. Example cont. • b. digits cannot be repeated? • c. digits cannot be repeated and the number must be even?

10. The Complement Principle • If A is a subset of a universal set U, then: n(A) = n(U) – n()

11. Example: • Find the number of 3-digit numbers containing at least one digit 8.