Counting Principle and Probability

The fundamental counting principle

• Tells you the probability of an event.

• Tells you how many different possible outcomes there are.

• Helps you convert a fraction into a percent.

• Shows you the outcomes that don't appear in a tree diagram.

If I use the fundamental counting principle, how many different outcomes are there if I flip 4 different 2-sided coins?

• 8, because 4 x 2 = 8

• 6, because 4 + 2 = 6

• 16, because 2 x 2 x 2 x 2 = 16

• 256, because 4 x 4 x 4 x 4 = 256

Using the fundamental counting principle, how many different outcomes could you have if you rolled 4 different six-sided dice?

• 1,296, because 6 x 6 x 6 x 6 = 1,296

• 24, because 6 x 4 = 24

• 36, because it 6 x 6 = 36

• 360, because 6 x 5 x 4 x 3 = 360

Use the fundamental counting principle to find the total number of outcomes (or outfits) if you have a suitcase with 5 different shirts, 3 types of jackets, 2 different pants, and 3 types of shoes.

• 13, because 5 + 3 + 2 + 3 = 13

• 120, because 5 x 4 x 3 x 2 =120

• 90, because 5 x 3 x 2 x 3 = 90

• 52, because (5 + 3 + 2 + 3) x 4 = 52

Use the fundamental counting principle to find the total number of possible outcomes for my smartphone lock screen if I have to use 4 different numbers from 0-9, and I can reuse numbers. (For instance, I could use "1111")

• 6,561, because 9 x 9 x 9 x 9 = 6,561

• 5,040, because 10 x 9 x 8 x 7 = 5,040

• 10,000, because 10 x 10 x 10 x 10 = 10,000

• 1111, because that's the number in the problem.

Using the fundamental counting principle, how many different possible codes are there to lock my iPhone if I use 4 digits, each digit has to be a number from 0-9, and I cannot use the same number twice? (For instance, I could use codes like 1234)

• 5,040, because 10 x 9 x 8 x 7 = 5,040

• 1234, because that's the number in the problem

• 10,000 because 10 x 10 x 10 x 10 = 10,000

• 40, because 10 + 10 + 10 + 10 = 40

Use the fundamental counting principle to find all the different types of 1-topping pizzas are possible if you could choose from 4 types of crusts, 5 types of cheese, 2 sauces, and 10 types of toppings.

• 21, because 4 + 5 + 2 + 10 = 21

• 84, because (4 + 5 + 2 + 10) x 4 = 84

• 1, because it's a 1-topping pizza

• 400, because 4 x 5 x 2 x 10 = 400

Use the fundamental counting principle to find the total number of possible outcomes for lunch if you can choose from 4 types of sides, 6 types of main entree, and 12 types of drinks.

• 3, because you are choosing 1 side, 1 entree, and 1 drink.

• 22, because 4 + 6 + 12 = 22

• 66, because (4 + 6 + 12) x 3 = 66

• 288, because 4 x 6 x 12 = 288

Use the fundamental counting principle to find the total number of outcomes for an election if there are 3 different positions and there are 8 people running for the positions. Remember, one position can only be held by one person.

• 24, because 3 x 8 = 24

• 81, because 3 x 3 x 3 x 3 = 81

• 4,096, because 8 x 8 x 8 x 8 = 4,096

• 1,680, because 8 x 7 x 6 x 5

Use the fundamental counting principle to find the total number of different outcomes for a car license plate if you need 3 different letters and 3 different 1-digit numbers. You can reuse numbers and letters (for instance, AAA-111 is possible). There are 26 letters in the alphabet, 1-digit numbers are 0-9.

• 6, because there are six different spaces in the license plate.

• 17,576,000, because 26 x 26 x 26 x 10 x 10 x 10

• 12,812,904, because 26 x 26 x 26 x 9 x 9 x 9

• 9, because 3 x 3