Concept 1 · free preview
The basic counting principle (AND / OR)
Understand
This is the base of the whole chapter: if a job is done in several steps, multiply the ways for each step; if it is done either one way or another, add.
Rule
AND (this step, then that step too): m × n
OR (either this or that — not both): m + n
Repetition allowed: every place has all the choices → n × n × … (r times) = nʳ
Repetition not allowed: each next place has one choice fewer → n × (n − 1) × …
Example
3 shirts and 4 trousers → one shirt AND one trouser = 3 × 4 = 12 ways. To reach Delhi by one of 3 buses or one of 2 trains → 3 + 2 = 5 ways.
Example
3-digit codes from the digits 1 to 5: Repetition allowed → 5 × 5 × 5 = 125 No repetition → 5 × 4 × 3 = 60
Remember
‘Send each item somewhere’ questions: each item gets the choices. 4 letters, 3 letter boxes → each letter has 3 choices → 3⁴ = 81. ‘At least one repeated’ → total − ‘no repeats’. 3-digit numbers: 900 − (9 × 9 × 8) = 900 − 648 = 252.
Common mistake
Flipping the power in letters-and-boxes: 4 letters in 3 boxes is 3⁴ = 81, not 4³ = 64 (the base is each item’s number of choices).
Check yourself
4 students each pick one of 5 clubs (a club can take many students). How many ways in all?