Mathematics from zero
The counting principle
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.
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.