Combinations and the Fundamental Counting Principle

The Fundamental Counting Principle determines the number of combinations one can make with multiple sets.

For example, an ice cream store has many flavors of ice cream and many types of toppings. The flavors are elements in one set while the toppings are elements in another set. If you are allowed to choose one element from each set, you can determine how many combinations you can make by simply multiplying the number of elements in each set together.

For example, let’s say an ice cream store has only 2 flavors, chocolate and vanilla, and 2 toppings, sprinkles and gummies. How many combinations of ice cream can you get?

Assuming you only eat ice cream with toppings, there are 4 combinations. You can choose chocolate ice cream with sprinkles (1) or with gummies (2), and you can choose vanilla ice cream with sprinkles (3) or with gummies (4). This idea is known as the Fundamental Counting Principle. 

According to the Fundamental Counting Principle, the number of possible combinations of data from independent sets can be determined by finding the product of the number of elements from each set.

Formula for Fundamental Counting Principle

M × N = C

M is the number of elements from one independent set.

N is the number of elements from another independent set.

C is the number of possible combinations from the independent sets M and N. 

In the ice cream problem, set M had 2 elements (chocolate and vanilla), and set N had 2 elements (sprinkles and gummies). So the number of combinations (C) is 4. Take a look at another example:

Problem

An ice cream store has 5 flavors and 6 toppings. If you must choose one flavor and one topping, how many combinations can you make?

Solution

1. The ice cream flavors are a set and the toppings are another set.
2. Find the product of 5 flavors and 6 toppings.
3. 5 × 6 =  30 combinations 

Making More than One Choice

The Fundamental Counting Principle also works when you are making more than one choice from independent sets, but you must pay attention to how you are making those choices.

• If you can repeat choices made, then you count each set the number of times you are making choices.
• If you cannot repeat choices made, then you count each set the number of times you are making choices and deduct an element from the set each time.

It’s like probability where you pull a marble out of a bag and put it back versus you pull a marble out of a bag and do not return it. Here is another example, with and without repeating choices:

Problem

An ice cream store has 10 flavors and 5 toppings. If you must choose two flavors and two toppings, how many combinations can you make?

If you can repeat choices:
10 × 10 × 5 × 5 =  2,500 combinations 
If you cannot repeat choices:
10 × 9 × 5 × 4 =  1,800 combinations 

Keep this principle in mind when you are asked to find the combinations of sets. The Fundamental Counting Principle underlies other aspects of probability, including combinations and permutations.

Practice Problems

1. An ice cream store has 8 flavors and 11 toppings. If you must choose 1 each, how many combinations can be made?
2. A hotdog-stand has 4 types of wieners and 10 types of toppings. If you must choose 1 wiener and 3 different toppings, how many combinations are possible?
3. Your closet contains 3 pairs of jeans, 4 shirts, 5 pairs of different colored socks, and 2 pairs of shoes. If you select 1 article of clothing from each set, how many combinations are possible?
4. In a restaurant you have 3 appetizers, 7 entrees, and 4 side dishes to choose from. How many possible combinations are there if you choose 1 item from each set?

Answers to Practice Problems

1. 8 × 11 = 88
2. 4 × 10 × 9 × 8 = 2,880
3. 3 × 4 × 5 × 2 = 120
4. 3 × 7 × 4 = 84