Suppose you are in a small restaurant and are ready to order a soup, a main course, and a beverage.

(Unfortunately, you will need to go somewhere else for dessert!) The restaurant has 2 different soups, 3

different main courses, and 5 different beverages. If a meal consists of one item from each of the three

categories, how many different meals are possible?

2x3x5 = 30 ways to select one item from each of the three categories.

Counting Methods – Part 1

Use basic counting methods to determine the number of ways of getting a sequence of events.

Practical Situations

1. Selecting a batting order for a team of baseball players

2. Selecting a set of courses to take at a college

3. Selecting a soup, dessert, and beverage when having dinner at a restaurant

Let’s visit a better restaurant!!!• Suppose this restaurant has 4 different soups, 12

different main courses, 10 different desserts, and 8 different beverages. (Of course, you will have the opportunity to select a dessert after finishing the main course!) If a meal consists of one item from each of the four different categories, how many different meals are possible?

4x12x10x8 = 3840

Factorial!

• 5! Read as “five factorial”• 5 x 4 x 3 x 2 x 1 = 120• The product involves starting with a

positive integer and multiplying by each decreasing consecutive integer, until reaching the number 1.

• Factorials are not defined for fractions or decimals.

What is 0!?

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4!=4x3x2x1=243!=3x2x1=62!=2x1=21!=1

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Notice:4!÷4 =3!3!÷3 =2!2!÷2 =1!1!÷1=0!

1!Since =1, 1!then ÷1=1÷1=1

0! is 1

Six people are to be assigned to 6 different seats. In how many ways can this be done?Seats 1 2 3 4 5 6

Choices 6 5 4 3 2 1

6! = 720To enter 6! on calculator:•MATH•Arrow right to PRB•Choose 4

Ten people wish to line up for a photograph. How many different arrangements are possible?

10! = 3,628,800 different arrangements

A high school bowling club consists of 8 members. A president, vice president and a

treasurer are to be chosen. If 3 different people will be selected, in how many ways

can this be done?

Since there are only 3 available positions:8(7)(6)=336 ways can be done

4!

• What is another way to express 4!?

Properties of factorials:

• Factorials cannot be used for decimals, fractions or any negative numbers.

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12

⎛

⎝ ⎜

⎞

⎠ ⎟!, (2.42)!, (-6)! .have no meaning

Properties of factorials:

• The product of two factorial numbers is not equal to the factorial of their product or any arithmetic operation.

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2!+4!≠6!, 3! 5!x ≠15!, 20!÷10!≠2!

Properties of factorials:

• If n is any positive integer, then n! x(n+1)=(n+1)!

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12!×13 =13!12!÷12 =11!

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or (n+1)!÷(n+1) = !n