Count choices made one after another with the multiplication principle, separate cases with the addition principle, and meet the factorial.
3 lessons, about 1 h 15 min3 simulations10 problems4 practice questionsPython in every lesson
Created by StemPy LabWhen a choice is made in stages, one after another, multiply the numbers of choices at each stage. With ways for the first stage and ways for the second, there are outcomes. A tree diagram shows why. When repetition is allowed, every stage keeps all its choices. When it is not, each stage has one choice fewer than the stage before.
When a task can be done in one way OR another, and the two sets of ways do not overlap, add the counts: this is the addition principle. Choices that follow each other (AND) multiply; separate cases (OR) add. Split a hard count into cases that cannot overlap, count each with the multiplication principle, and add. When a place has a rule, such as an odd last digit, fill that place first.
The product of the whole numbers from to is written and read "n factorial": . Each factorial is the one before times the new number, , and that rule with shows why . Ratios of factorials cancel down to a short product, and a product of numbers counting down is a ratio of factorials.
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