Fundamental Counting Principle

Algebra 2 · Axiom Academy

LESSON Fundamental Counting Principle Count a sequence of independent choices by multiplying the options at each stage — and see why you multiply instead of add. Make a sequence of independent choices — a first with n_1 options, a second with n_2 , on up to an n_k -th with n_k . Draw every possibility as a branching tree: each new stage multiplies the number of branches, so the number of leaves at the far end is the product . Watch the running count grow one stage at a time. The rule for k independent stages The tree below: 2, then 3, then 2 options Why not just add the options? Because the stages are not alternatives you choose between — you take one option from each . Every option in the first stage pairs with every option in the second, and those pairings fill a grid. 3. When Choices Repeat — and When They Don't The per-stage counts hinge on one question: can an option be reused? If a choice can repeat (like a digit in a PIN), every stage keeps its full count. If items get used up (like people placed in a line), each stage has one fewer than the last. A 3-symbol code from 5 symbols, repeats OK: . The pool never shrinks. Arrange 3 of 5 distinct items: . Each pick removes one option. A plate is three letters followed by three digits, and any symbol may repeat. Substitute each stage's count and multiply: 26 choices for each of the three letters, 10 for each of the three digits. A 5-digit passcode, each position a digit 0 – 9 (repeats allowed): codes.

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