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Now · Week map
Deep study for Quiz 2 week 5. Counting stories reduce to OR (add) vs AND (multiply) — identify structure before computing.
Week map
Outcome listing → addition rule (mutually exclusive) → multiplication rule (sequential stages) → factorial arrangements → tree diagrams.
Counting notation
- n! → “n factorial” → product n×(n−1)×⋯×1 → ways to arrange n distinct items in a line.
- 0!=1 → empty arrangement convention.
- Mutually exclusive → events cannot happen together → use addition.
- Independent stages → choices in sequence → multiply counts per stage.
Mini-example: 4!=4×3×2×1=24 orderings of four distinct books.
Addition rule (OR)
If task A can be done in n1 ways and task B in n2 ways, and they are disjoint (only A or only B):
ways=n1+n2
Example: pick soup or salad (not both) from menus of 3 soups and 4 salads → 3+4=7.
Multiplication rule (AND)
If stage 1 has n1 options and stage 2 has n2 options (independent sequential choices):
ways=n1×n2
Example: 3 shirts and 2 pants → 3×2=6 outfits.
Pattern families
Easy — OR count
Disjoint options: add counts. Identify that choices exclude each other.
Medium — AND stages
Multiply choices per stage. Tree diagram: branches multiply along a path.
Hard — Factorial setup
Arrange all n distinct objects in order → n!. Distinguish arrangement (order matters) from selection (week 6). Spot when objects are not all distinct (preview: divide later).
Worked mini-examples
Example 1 — Addition.
Travel by bus (4 routes) or metro (2 routes), not both → 4+2=6.
Example 2 — Multiplication.
PIN with 2 letters (26 each) then 3 digits (10 each) → 262×103 (if repetition allowed — state assumption).
Example 3 — Factorial.
Arrange 3 students in a row → 3!=6.
Example 4 — Tree.
Choice A/B then choice 1/2/3 → 2×3=6 paths.
Example 5 — Not disjoint trap.
Red shirt or blue shirt from 3 red and 2 blue → add works. “Shirt or pants” from same outfit pick may need careful reading.
Traps
- Multiplying when outcomes overlap (not mutually exclusive).
- Adding when stages should multiply.
- Forgetting 0!=1.
- Treating identical objects as distinct without reason.
- Double-counting paths on tree when branches reconverge.
Diagnostic (try yourself)
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Menu: 5 vegetarian or 7 non-vegetarian mains (pick exactly one). How many choices?
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Lock with 3 dials, 10 digits each. How many combinations (repetition allowed)?
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In how many ways can 4 distinct trophies line up on a shelf?
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A task has two sequential choices: 3 then 5 options. Total paths?
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Why is 0! defined as 1? (One-sentence intuition.)