One, two, three, we’re… Counting

Learning Objectives Fundamental Counting Principle - Sum Rule - Product Rule Counting without and with Restrictions Factorial Formula and Notation

Basic Counting Principles Counting problems are of the following kind: “How many different 8-letter passwords are there?” “How many possible ways are there to pick 11 soccer players out of a 20-player team?”

Basic Counting Principles Example: ISF Academy will award a free computer to either a Math teacher or a student. How many different choices are there, if there are 530 students and 6 Math teachers? There are 530 + 6 = 536 choices. The sum rule: If a task can be done in n1 ways and a second task in n2 ways, and if these two tasks cannot be done at the same time, then there are n1 + n2 ways to do either task. Key Word: “Or”

Basic Counting Principles Example: Tiger has 8 red pens, 3 blue pens and 9 green pens. How many different pens can he use? Solution: Since he can choose either red, blue or green, this is the sum rule. He can use 8 + 3 + 9 = 20 different pens.

Basic Counting Principles Example: Mr Loo has 5 shirts and 3 pants. How many different outfits can he wear? Solution: 5 x 3 = 15. But why?

Basic Counting Principles Example: How many different license plates are there that containing exactly three English letters ? Solution: There are 26 possibilities to pick the first letter, then 26 possibilities for the second one, and 26 for the last one. So there are 26 x 26 x 26 = 17576 different license plates.

Basic Counting Principles The product rule: Suppose that a procedure can be broken down into two successive tasks. If there are n1 ways to do the first task and n2 ways to do the second task after the first task has been done, then there are n1 x n2 ways to do the procedure. Key word: “AND”

Basic Counting Principles The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. In how many ways can a person order a two-course meal? Solution: Choosing from one of 6 appetizers and one of 14 main courses, the total number of two-course meals is: 6 x 14 = 84

Counting without restrictions I want to line 10 students in a line. How many ways can I arrange these students? Solution: 10x9x8x7x6x5x4x3x2x1 = 3,628,800 ways

Factorials For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n !. Example: 5! = 5 x 4 x 3 x 2 x 1 = 120

Examples: Evaluate each expression. a) 4! b) (4 – 1)! Solution: 4 x 3 x 2 x 1 = 24 3! = 3 x 2 x 1 = 6

Counting with Restrictions How many two-digit numbers that do not contain repeated digits can be made from the numbers 0, 1, 2, 3, 4 and 5? Solution: There are only 5 ways to pick the first numbers and 5 ways to pick the 2nd number. Therefore by using the product rule, There are 5 x 5 = 25 numbers

Counting with Restrictions Some license plates consist of 3 letters followed by 3 numbers. How many different license plates are possible if there are no restrictions if the letters are different. Solution: a) 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000 b) 26 x 25 x 24 x 10 x 10 x 10 = 15,600,000