Counting in cases

When the ways split into separate cases that cannot happen together, count each case and add. 'And then' multiplies; 'or' between separate cases adds.

Example: to reach the stadium a team can take one of 3 buses or one of 2 trains, giving 3 + 2 = 5 ways. Taking a bus and then a train would give 3 × 2 = 6 ways.

Flag signals: 4 house flags of different colours are hung one below another on a pole, and a signal uses at least 2 flags. Two flags: 4 × 3 = 12. Three flags: 4 × 3 × 2 = 24. Four flags: 4 × 3 × 2 × 1 = 24. Total 12 + 24 + 24 = 60 signals.

Allowing single-flag signals as well adds 4 more, so 'at least one flag' gives 64.

Repetition changes the count. Three-digit numbers from the digits 1, 2, 3, 4, 5: with repetition, 5 × 5 × 5 = 125; without repetition, 5 × 4 × 3 = 60.