Concept 7 of 7
Counting Principle: Multiplying Choices
Today's big idea
Count the options in each slot, then multiply those counts straight across.
Repeats allowed or not changes the count, so the question always says which.
How it works
👀 What's going on?
Two independent choices in a row, and someone asks how many ways there are altogether. Listing them works for small numbers and collapses for large ones.
💡 Here's the trick
Look at the slots above. Every choice in the first slot can be followed by every choice in the second, so the counts multiply rather than add.
🪄 How to do it
- 1.Count the slots above, one for each independent choice.
- 2.Write the number of options into each slot.
- 3.Multiply the numbers straight across.
✍️ Try it
A canteen has 6 mains and 4 puddings; how many two-course meals? Then: how many 3-letter codes use only A, B, C and D, with letters allowed to repeat?
Each main pairs with each pudding, giving 24 meals. Each of the three positions has 4 choices, so there are 4 times 4 times 4, which is 64 codes.
⚠️ Watch out
You might be thinking repeats are always allowed. They are not. If a letter cannot be reused, the second slot has one fewer choice than the first, and the count drops.
🌟 Why this matters
Passwords, number plates, timetables and menus are all counted this way, which is why a longer password is so much harder to guess.
Exam tip: Ask whether repeats are allowed before you multiply; the question always says, and the answer changes.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A wardrobe has 5 shirts and 4 ties. How many shirt-and-tie outfits are possible?”
- 1
Two independent choices
One from shirts, one from ties. 🧮
- 2
Apply counting principle
5 × 4
- 3
Multiply (never add)
20
- 4
Answer ✓
20 outfits.
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
When choices are independent, MULTIPLY the number of options for each.
COMBINATIONS · ROUND 1 OF 3
4 T-shirts and 3 shorts. How many different OUTFITS?
Quick check
✅ VERIFICATION
How many 2-letter codes can be made from P, Q, R and S if a letter may NOT be repeated?
⚡ Finish the activity above first to answer.
Keep going!