How do you use the multiplication principle of counting?

The core rule of multi-stage choices

The multiplication principle of counting states that if an experiment occurs in consecutive stages, you find the total number of outcomes by multiplying the number of choices available at each stage. For example, if you choose one shirt from 4 options and one pair of pants from 3 options, you have 4 times 3, or 12 distinct outfits.

This principle forms the foundation of probability. Before you can calculate the likelihood of an event, you must know the size of the total sample space. Multiplying stage options gives you that count without listing every combination by hand.

Repetition allowed versus repetition not allowed

When choices repeat, the number of options stays constant at each stage. Consider a three-digit lock using the digits 0 through 9. If you may repeat digits, each slot has 10 options, giving 10 times 10 times 10, which equals 1,000 possible codes.

When items cannot repeat, the available choices decrease by one with each step. If you create a three-digit code without repeating any digit, the first slot has 10 options, the second has 9, and the third has 8. That gives 10 times 9 times 8, or 720 possible codes.

Arranging a full set of objects in a row follows this decreasing pattern. Arranging 5 books on a shelf gives 5 times 4 times 3 times 2 times 1, which equals 120 unique arrangements.

Connecting counts to classical probability

Once you know the total number of outcomes, you can calculate classical probability. Divide the number of favorable outcomes by the total count in the sample space.

Suppose a computer creates a random 4-letter password using the letters A, B, C, D, and E without repeats. The total possible passwords equal 5 times 4 times 3 times 2, which is 120. If only one specific order unlocks the file, the probability of guessing it on the first attempt is 1 over 120.