The multiplication principle

Counting techniques find how many ways something can happen without writing out every possibility, which quickly becomes impossible as the numbers grow.

Multiplication principle: if one job can be done in m ways and, after it, a second job can be done in n ways, the two jobs in that order can be done in m × n ways. The rule extends to three or more jobs: m × n × p, and so on.

Sports-day kit: a runner picks one of 3 T-shirt colours, one of 2 caps and one of 2 water bottles. The number of different kits is 3 × 2 × 2 = 12, which a tree diagram with 3, then 6, then 12 branch ends confirms.

Locker lock: 3 dials, each showing a digit 0 to 9. If digits may repeat, there are 10 × 10 × 10 = 1000 codes. If no digit may repeat, the second dial has only 9 choices left and the third 8, giving 10 × 9 × 8 = 720 codes.

Fill the most restricted place first. For two-digit even numbers from the digits 1 to 6 with repetition allowed, the units place must be 2, 4 or 6 (3 ways) and the tens place can be any of the 6 digits, so there are 3 × 6 = 18 such numbers.

A coin tossed 4 times has 2 × 2 × 2 × 2 = 16 possible outcome sequences, because each toss has 2 results whatever happened before.