The counting principle
When you make several independent choices in a row, the number of possible outcomes is the choices multiplied together — count without ever listing.
You own 3 shirts and 2 pairs of trousers. How many different outfits can you make? You could list them all — but there is a faster way. Pick a shirt 3 ways, then trousers 2 ways: 3 times 2 is 6 outfits, counted without writing a single one down.
After this lesson you can state the counting principle, apply it to two or more choices made in a row, and explain why counting outcomes means multiplying.
The counting principle: multiply the number of options at each choice. If one
choice can be made in A ways and a second choice in B ways, then making both — one
after the other — can happen in A × B ways. Three shirts and two trousers give
3 × 2 = 6 outfits.
The principle extends to as many choices as you like. Add a third choice with C
options and the total becomes A × B × C. Each new independent choice multiplies the
count again. A breakfast of one cereal from 4, one fruit from 3, and one drink from 2
has 4 × 3 × 2 = 24 possibilities.
The principle works when the choices are independent. Independent means each choice’s options do not depend on what you picked before — every shirt still pairs with every pair of trousers. When that holds, you simply multiply. If a later choice’s options shrink because of an earlier one, the plain multiply needs adjusting — the next lesson handles that case.
This is why combinatorics is built on multiplication. Counting possibilities by listing them works for tiny problems and collapses for real ones — 10 choices of 10 options would be ten billion outcomes. The counting principle replaces the impossible list with one multiplication. Every counting technique ahead grows from this one idea. When you hit a counting problem, ask yourself: how many choices are there, and how many options at each? Then multiply — the list writes itself.
A café meal is one main from 4, one dessert from 3, and one drink from 2. How many different meals are possible?
There are three choices made in a row. The mains offer 4 options, the desserts 3,
the drinks 2. The choices are independent — any main goes with any dessert and any
drink.
Apply the counting principle: multiply the options at each choice. 4 × 3 = 12, then
12 × 2 = 24.
There are 24 possible meals. Listing all 24 would be slow and error-prone; the
multiplication gives the count in one line.
▸Why this works
Why multiply rather than add? Because each option of the first choice opens up a
fresh full set of the second choice’s options. Shirt 1 has all 2 trouser options;
so does shirt 2; so does shirt 3. That is 2, three times over — 2 + 2 + 2, which is
3 × 2. Multiplication is repeated addition, and that is exactly the shape of
counting nested choices.
▸Common mistake
A common mistake is adding the options instead of multiplying — saying 3 shirts and 2 trousers give “5”. Adding answers a different question: “how many garments do I own?” The counting principle answers “how many combinations can I form?”, and combinations multiply.
3 shirts and 2 pairs of trousers. How many outfits? Type the count.
4 mains and 5 desserts. How many main-and-dessert meals? Type the count.
Flip a coin 3 times — each flip has 2 outcomes. How many outcome sequences? Type the count.
You make one choice from 10 options. How many outcomes? Type the count.
5 hats and 3 scarves. How many hat-and-scarf pairings? Type the count.
You make a choice with 4 options, then a separate choice with 3 options. How many combined outcomes are there?
The counting principle says: when you make several independent choices in a row,
multiply the number of options at each one to get the total number of outcomes. It
extends to any number of choices — A × B × C and so on. It works whenever the
choices are independent. Counting possibilities means multiplying, because each early
option opens a full fresh set of the later options. Now when you meet a counting
problem — passwords, menus, routes — your first move is to identify the choices and
multiply, not to list.
Practice
Start at the top. Tasks go easiest → hardest: recall a fact, apply it to a case, then a senior-level stretch. Open one, attempt it, then reveal.
Something unclear?
Ask a question about this lesson. Questions are anonymous and go straight to the author to make the lesson better.
Apply this
Put this lesson to work on a real build.