Fundamental Principles of Counting

Learning Outcome

  • Identify and use the Addition Principle of Counting
  • Identify and use the Multiplication Principle of Counting (Fundamental Counting Principle)
  • Find the number of permutations of n distinct objects

The Addition Principle of Counting

The Addition Principle states that if two sets of items are distinct from one another (there is no overlapping), then the sum of the union of the sets is obtained by adding the sum of each set together. This probably appears to be a rather straightforward statement, and it is. But without it, counting anything would not be possible. Counting begins here with examples like the following that include mutually exclusive sets.

Ex. On a certain used car lot, there are [latex]9[/latex] blue cars and [latex]12[/latex] green cars. There are no cars that are both blue and green at the same time. To determine the total number of cars, we do not have to count them individually. We can simply sum up the sets of cars: [latex]9 + 12 = 21[/latex] total cars.

Ex. In a certain college algebra class, there are 18 freshmen and 7 sophomores. [latex]18 + 7 = 25[/latex] students.

Ex. We can use mathematical notation to illustrate the principle. Let the probability of a certain event [latex]U[/latex] be [latex].09[/latex] and the probability of a separate, mutually exclusive event [latex]V[/latex] be [latex]0.34[/latex]. Then the sum of the two probabilities can be given by [latex]P\left(U\right)[/latex] or [latex]P\left(V\right)[/latex]. That is, the probability of either event happening is [latex]P\left(P \text{ or } V\right)=0.09+.34=0.43[/latex].

A General Note: The Addition Principle

According to the Addition Principle, if one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways.

The Multiplication Principle of Counting

The Multiplication Principle, also called the Fundamental Counting Principle, states that if there are so many ways one event can occur after another has already occurred, the total number of ways the two can occur together can be found by multiplying. A classic example presents the choice made at a lunch counter.

Ex. A certain lunch special permits one sandwich, two sides, and a drink. There are five sandwiches, seven sides, and four drinks to choose from. How many possible lunch combinations can be made? The Multiplication Principle states that there are [latex]5\ast7\ast4=140[/latex] possible combinations.

A General Note: The Multiplication Principle

According to the Multiplication Principle, if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways. This is also known as the Fundamental Counting Principle.

Find the Number of Permutations of n Distinct Objects

we can use the Multiplication Principle to find the number of ways to arrange items or people in a specific order. For example, in a race with [latex]6[/latex] contestants, how many possible orders are there (all things being equal) in which they may cross the finish line? The Multiplication Principle states that we may multiply together the possible number of contestants available to fill each position.

Before the first person crosses the finish line, there are [latex]9[/latex] possibilities of who can take that spot. But after the winner crosses the line, there are only [latex]8[/latex] remaining possibilities to take second place, and once that position has been claimed, there are only [latex]7[/latex] remaining, and so on until only [latex]1[/latex] person is left to take the final position. Using the Multiplication Principle, we can take the product to find that there are [latex]9\ast8\ast7\ast6\ast5\ast4\ast3\ast2\ast=362,880[/latex] possible ways.

If there are only four positions available to take out of the nine runners, how many possible ways can the four positions be filled? Take the product of the possibilities until the positions have been filled. There are [latex]9\ast8\ast7\ast6=3024[/latex] possible ways for four of the nine racers to finish in the top four positions.

How To: Given [latex]n[/latex] distinct options, determine how many permutations there are.

  1. Determine how many options there are for the first situation.
  2. Determine how many options are left for the second situation.
  3. Continue until all of the spots are filled.
  4. Multiply the numbers together.