Statistics I · Week 5 — Counting
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2026-08-16T00:00:00.000Z
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addition vs multiplication rule, factorials — concepts, pattern families, and traps for Quiz 2 week 5. # Week 5 — counting > **Quiz 2 scope:** Weeks 1–8 per IITM May 2026 foundation courses.
Week 5 — counting
Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.
Part of the Quiz 2 prep system%20%C2%B7%20%5BWeeks%201%E2%80%938%20index%5D(.%2Fmay-2026-stats-quiz-2-weeks-1-8-prep) · Pattern atlas · Formula chains.
Week map
Events → addition OR → multiplication AND → factorial arrangements
Classify → Represent → Execute → Trap-check
- Recognize: Ask: When add vs multiply counts?
- Procedure: Disjoint options: add counts. Example: pick meal OR drink from separate disjoint menus.
- Variations / traps: Watch for: Multiplying when outcomes overlap (not disjoint).
Formula chain (compressed)
addition rule (or) → multiplication rule (and) → factorial n!.
- Addition rule —
|A∪B| = |A|+|B|−|A∩B|— either A or B - Multiplication —
k choices × m choices— sequential independent stages - Factorial —
n! = n×(n−1)×...×1— arrange n distinct - Disjoint —
A∩B = ∅ → |A∪B|=|A|+|B|— no overlap - With/without replacement —
denominator shrinks?— changes counts
Open interactive formula desk · Week 5 tab.
Deep study
Statistics I · Week 5 — Counting rules
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
- → “ factorial” → product → ways to arrange distinct items in a line.
- → empty arrangement convention.
- Mutually exclusive → events cannot happen together → use addition.
- Independent stages → choices in sequence → multiply counts per stage.
Mini-example: orderings of four distinct books.
Addition rule (OR)
If task A can be done in ways and task B in ways, and they are disjoint (only A or only B):
Example: pick soup or salad (not both) from menus of 3 soups and 4 salads → .
Multiplication rule (AND)
If stage 1 has options and stage 2 has options (independent sequential choices):
Example: 3 shirts and 2 pants → 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 distinct objects in order → . 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 → .
Example 2 — Multiplication.
PIN with 2 letters (26 each) then 3 digits (10 each) → (if repetition allowed — state assumption).
Example 3 — Factorial.
Arrange 3 students in a row → .
Example 4 — Tree.
Choice A/B then choice 1/2/3 → 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 .
- Treating identical objects as distinct without reason.
- Double-counting paths on tree when branches reconverge.
Diagnostic (try yourself)
-
Menu: 5 vegetarian or 7 non-vegetarian mains (pick exactly one). How many choices?
-
Lock with 3 dials, 10 digits each. How many combinations (repetition allowed)?
-
In how many ways can 4 distinct trophies line up on a shelf?
-
A task has two sequential choices: 3 then 5 options. Total paths?
-
Why is defined as 1? (One-sentence intuition.)
ChatGPT prep archive
Archived import for extra depth — complements the notes above, not official IITM material.
Core concepts
- Addition rule: mutually exclusive outcomes: ways = n1 + n2.
- Multiplication rule: sequential independent stages: ways = n1 × n2.
- Factorial n!: permutations of n distinct items: n!.
- Identify: OR vs AND story structure before computing.
Notation & vocabulary
| Rule | When |
|---|---|
| + | either A or B, not both |
| × | A then B stages |
| n! | all orderings of n distinct |
Pattern families
Easy — OR count
Disjoint options: add counts. Example: pick meal OR drink from separate disjoint menus.
Medium — AND stages
Shirt then pants: multiply choices per stage. Tree diagram multiplies along paths.
Hard — Factorial setup
Arrange all in line: n!. Identify whether problem is arrangement (order) vs selection (later week).
Drill these on the pattern atlas — filter to week 5.
Traps
- Multiplying when outcomes overlap (not disjoint).
- Adding when stages should multiply.
- 0! assumed 1 forgotten.
- Treat same objects as distinct without reason.
Retrieval prompts
- When add vs multiply counts?
- What does n! count?
- Value of 0!
Practice loop
- Read Deep study (if present) or core concepts once.
- Recite the formula chain without looking.
- Open one easy pattern on the interactive atlas for week 5.
- Attempt without solutions; mark studied after an honest try.
- Say one trap aloud before closing the tab.