Counting Cases
“Happening in a row: multiply. One of two: add.”
The formula
N = m × nHow to read it: if one event can happen m ways and then another can happen n ways, the two happening in a row can occur m×n ways
- m
- — the number of ways the first event can happen
- n
- — the number of ways the next event can happen
- m × n
- — the total number of ways both happen in a row
The hook
'Counting cases' is really the skill of counting every possibility once — nothing missed, nothing double-counted.
In plain words
Things that can't happen together you ADD (the sum rule); things that happen one after another you MULTIPLY (the product rule).
The intuition
Picking an outfit: choose 1 of 3 shirts, then 1 of 2 pants. Each shirt branches into 2 pant choices, so 3×2 = 6 branches. It splits like a tree, so you multiply.
How it's built
The key test is 'and' versus 'or'. One then another in a row → multiply ('and'). Either this or that, only one → add ('or').
Example
Ways to wear one of 3 T-shirts with one of 2 pants: 2 pants for each shirt → 3×2 = 6 outfits.
Common mistake
Adding when it's really 'and', or multiplying when it's really 'or', goes wrong. Happening in a row → multiply; only one of the two → add.
Where it's used
The number of possible passwords, seating arrangements, tournament brackets, the denominator of a lottery probability — to find a probability, you first have to count the cases.
Where it came from
Counting possibilities systematically took off in the 1600s, when Pascal and Fermat cracked gambling problems and opened the door to probability theory.
Quick check
Tossing one coin and rolling one die at the same time, how many outcomes are possible?
- 8
- 12✓
- 36
- 2
Practice
How many ways are there to pick a set from 4 kinds of T-shirt and 3 kinds of hat?
Answer: 12
- Each shirt branches into 3 hat choices
- 4 × 3 = 12
Takeaway: Choices made one after another get multiplied.
Using the digits 1, 2, 3, how many two-digit numbers (like 12 or 33) can you make? (repeats allowed)
Answer: 9
- Tens place: 3 choices
- Units place: 3 choices (repeats allowed)
- 3 × 3 = 9
Takeaway: Multiply the number of choices at each place.
Rolling two dice, how many ordered pairs of faces are possible? On your notepad, write why it's multiplication.
Answer: undefined
- First die: 6 ways
- Second die: 6 ways
- Each face branches into 6, so 6 × 6 = 36
Takeaway: Events linked by 'and' have their counts multiplied.