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Mathematics · Ch 6 — Permutations and Combinations

Fundamental Principle of Counting

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Fundamental Principle of Counting

Many counting problems in real life boil down to a sequence of independent choices, each with a known number of possible outcomes. The Fundamental Principle of Counting (FPC) — also called the multiplication principle — gives the basic rule for combining such choices.

If an event can occur in mm different ways, and following it, a second event can occur in nn different ways (independently of how the first event occurred), then the two events together can occur in m×nm \times n different ways.

Why multiplication, not addition? Think of the two events as two independent stages of a single combined choice. For each of the mm ways the first stage can go, the second stage can still go in any of its nn ways — so the nn possibilities repeat for every one of the mm first-stage outcomes, giving mm groups of nn, i.e. m×nm \times n in total.

Example. A student has 3 different shirts and 2 different pairs of trousers. To choose an outfit, she picks one shirt (3 ways) and one pair of trousers (2 ways). Since these are two stages of a single combined choice, the total number of outfits is 3×2=63 \times 2 = 6 — not 3+2=53 + 2 = 5.

Extending to more than two stages

The principle extends naturally to any number of successive, independent stages. If an operation consists of kk stages that can be performed in m1,m2,…,mkm_1, m_2, \dots, m_k ways respectively, the entire operation can be performed in

m1×m2×⋯×mkm_1 \times m_2 \times \cdots \times m_k

ways. This is the working tool behind almost every formula developed later in this chapter — the formulas for nPr^{n}P_{r} and nCr^{n}C_{r} are both derived, not just stated, by breaking a selection or arrangement into a sequence of stages and applying this principle.

Watch out

Multiplication principle vs. addition principle

Use multiplication when a choice is made by combining stages that all happen together ("this stage and that stage"). Use addition when a choice is between mutually exclusive alternatives that cannot both happen ("this option or that option, but not both") — if an event can occur in mm ways or, alternatively (never together with the first), in nn ways, the event can occur in m+nm + n ways in total. Confusing the two — adding when the events are combined stages, or multiplying when they are exclusive alternatives — is one of the most common counting mistakes.

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