Systematic listing Worked Solutions — GCSE Maths

Fix the first coin as Heads

Keep the first coin on Heads and list the second coin's options: Heads then Tails.

Fix the first coin as Tails

Now the first coin is Tails, and again list the second coin's options.

List all outcomes

Working systematically gives all four outcomes with none missed or repeated.

Use $1$ as the tens digit

With $1$ first, the units can be $2$ or $3$ (not $1$, no repeats).

Use $2$, then $3$ as the tens digit

Repeat for tens digit $2$, then tens digit $3$.

List all six numbers

So there are six two-digit numbers.

Fix tea, vary the milk

For tea there are two choices: with milk or without.

Fix coffee, vary the milk

Same two choices for coffee.

List all four choices

So there are four possible drinks in total.

Start with $A$

Pair $A$ with each later letter.

Then start with $B$

$BA$ is the same as $AB$, so only $BC$ is new.

List the combinations

Since order does not matter, there are three combinations.

Pair Head with each dice number

Keep the coin on Heads and list all six dice results.

Pair Tail with each dice number

Now do the same with the coin on Tails.

Count them

Two coin results times six dice results gives $12$ outcomes.