Year 11 Methods — Addition & Multiplication Principles (10A)

Today's lesson

We're starting Counting Methods (Chapter 10). Counting is about working out how many ways something can happen — without writing out every possibility by hand. The two foundation rules are the addition principle and the multiplication principle. Get these right and the rest of the chapter follows.

Learning intentions

Use the multiplication principle for a sequence of choices made one after another (an "and" situation)
Use the addition principle for a choice between separate options (an "or" situation)
Draw a tree diagram to see why these rules work
Combine both rules in a single problem, and decide which one applies

Part 1 — The multiplication principle ("and")

Suppose you make a choice, then make a second choice, then a third. If the choices are made one after another ("this and then that"), you multiply the number of options at each stage.

Multiplication principle. If a first task can be done in $m$ ways, and for each of those a second task can be done in $n$ ways, then the two tasks together can be done in $m\times n$ ways. This extends to any number of stages.

Imagine a café meal: $3$ entrées, then $5$ mains, then $2$ desserts. A "meal" is one entrée and one main and one dessert, so we fill three boxes:

entrée

main

dessert

meals

So there are $3\times5\times2=30$ different meals. The "boxes" picture is worth keeping: one box per stage, write the number of options in each box, then multiply across.

📺 Walkthrough: a tree diagram for "2 shirts then 3 pairs of pants", showing why a sequence of choices multiplies to give $2\times3=6$ outfits.

Tree-diagram intuition. A tree diagram draws every choice as a branch. With $2$ shirts and $3$ pants, the first shirt splits into $3$ pants, and so does the second shirt — giving $3+3 = 2\times3 = 6$ end-branches. Each branch is one complete outfit. The multiplication principle is just a shortcut for counting the end-branches without drawing them all.

Start

Shirt A

→ Pants 1 • → Pants 2 • → Pants 3

Shirt B

→ Pants 1 • → Pants 2 • → Pants 3

EXAMPLE 1 — Multiplication principle

Old Victorian number plates had $3$ letters followed by $3$ digits ($0$–$9$), and letters or digits could repeat. How many different plates were possible?

There are six stages — three letter boxes then three digit boxes.
Each letter box has $26$ options; each digit box has $10$ options (repeats allowed).
Multiply across: $26\times26\times26\times10\times10\times10 = 26^3\times10^3$.
$26^3 = 17\,576$ and $10^3 = 1000$, so the total is $\boxed{17\,576\,000}$.

Now you try: A combination lock has $3$ dials, each showing $0$–$9$. How many settings are there? Answer: $10\times10\times10 = 1000$.

Part 2 — The addition principle ("or")

Sometimes you are choosing between separate, non-overlapping groups of options — "this or that". When the two cases can't both happen at once, you add the counts.

Addition principle. If one task can be done in $m$ ways and a separate task in $n$ ways, and the two cannot happen together, then doing one task or the other can be done in $m+n$ ways.

"And" → multiply. A sequence of choices, all made: outfit = shirt and pants. Use $\times$.

"Or" → add. One choice from separate options: travel by bus or by train. Use $+$.

EXAMPLE 2 — Addition principle

A lunch special lets you order one item. There are $5$ hot mains or $3$ light meals to choose from. How many different orders are possible?

You pick a hot main or a light meal — these are separate groups, and you choose only one item.
This is an "or" situation, so add: $5 + 3 = \boxed{8}$ different orders.

Watch out: if instead you chose a hot main and a light meal (two items), that would be "and" → $5\times3=15$. The word that decides it is or vs and.

Part 3 — Combining both principles

Many problems use both rules. The trick: break the problem into stages (multiply within a sequence) and into cases (add separate cases together).

EXAMPLE 3 — Both principles together

At a kiosk you choose a drink (from $4$ drinks) and a snack. The snack is either a pie (from $6$ pies) or a salad (from $3$ salads). How many drink-and-snack combinations are there?

The snack is a pie or a salad → add the snack options: $6 + 3 = 9$ snacks.
You take a drink and a snack → multiply: $4 \times 9 = \boxed{36}$ combinations.

Now you try: A meal deal is a drink ($3$ drinks) and a main, where the main is a burger ($4$ burgers) or a wrap ($2$ wraps). How many deals? Answer: $3\times(4+2)=18$.

Practice 3.1 — decide "and" (×) or "or" (+), then compute.

Toss a coin and roll a die. How many outcomes?
You may take one of $3$ bus routes or one of $2$ train lines to school. How many ways?
A 4-digit PIN ($0$–$9$, repeats allowed). How many PINs?
An outfit is one of $5$ shirts and one of $4$ ties. How many outfits?
You order a main (from $6$) and a dessert (from $4$). How many orders?
A code is $2$ letters then $4$ digits, repeats allowed. How many codes?

a) $2\times6=12$ b) $3+2=5$ c) $10^4=10\,000$ d) $5\times4=20$ e) $6\times4=24$ f) $26^2\times10^4=6\,760\,000$

Part 4 — Quick quiz (5 min)

Working program — Cambridge — Counting Methods Ex 10A

After the quiz, open Cambridge Methods 1&2 (Counting Methods, Chapter 10) and complete the set work:

Set work	Extension
Ex 10A — Q1, 3, 5, 7, 9	Try any "combined" questions and the chapter-review counting problems.

For every question, first decide: is each step an "and" (multiply) or an "or" (add)?

Exit ticket — write in your book

Before you pack up, write one sentence each:

When do you multiply the number of options, and when do you add?
Give one everyday example of an "and" situation and one of an "or" situation.
How does a tree diagram show why the multiplication principle works?