The Counting Principle (Multiplication Rule)

The fundamental counting principle

If a process has $k$ independent stages, and stage $i$ has $n_i$ choices, the total number of outcomes is the product: $\text{Total} = n_1 \cdot n_2 \cdot n_3 \cdots n_k$

"Independent" here means the choices at each stage do not depend on earlier stages — shirt 1 is still available no matter which pants you pick.

This is the engine behind nearly every counting problem in combinatorics.

Step-by-step solution

Stage 1 — Pick a shirt: 4 shirts → 4 choices. Stage 2 — Pick pants: 3 pants → 3 choices. Stage 3 — Pick shoes: 2 pairs → 2 choices.

Step 1 — Identify the stages and counts: three independent stages with counts $(4, 3, 2)$.

Step 2 — Multiply: $\text{Total outfits} = 4 \cdot 3 \cdot 2 = \boxed{24}$

Step 3 — Verify by drawing the tree. Root has 4 shirt-branches; each shirt-branch splits into 3 pants-branches (12 so far); each pants-branch splits into 2 shoe-branches. Leaves = $4 \cdot 3 \cdot 2 = 24$. ✓

Why the rule works

Each outfit corresponds to a unique root-to-leaf path in the tree. The branching factor at each level multiplies because, at each stage, every earlier choice is still compatible with every new choice.

Extensions

• Repetition allowed. PIN codes with 4 digits, each 0–9: $10 \cdot 10 \cdot 10 \cdot 10 = 10^4 = 10{,}000$.
• No repetition. License plates using each letter at most once: $26 \cdot 25 \cdot 24 \cdots$. The counts shrink by 1 at each stage.
• Restricted stages. A 4-digit PIN starting with an odd digit: first stage has 5 odd choices; the remaining three have 10 each → $5 \cdot 10 \cdot 10 \cdot 10 = 5000$.

Common mistakes

• Adding instead of multiplying. "Either-or" choices at a single stage add; stage-after-stage choices multiply. The outfit problem has stages in series, so multiply.
• Over-counting with dependencies. If stage 2 depends on stage 1 (e.g. "pick a matching tie"), you must count carefully — not every shirt has every tie available.
• Treating ordered arrangements as unordered. The counting principle gives ordered arrangements. If order shouldn't matter in your final answer, divide appropriately.

Try it in the visualization

Toggle each stage on or off, and adjust its option count. The branching tree redraws and the running product updates in real time, giving immediate visual feedback for why $4 \times 3 \times 2 = 24$.