Concept 4 of 7
Combinations: Count the Possibilities
Today's big idea
Count the choices in each group, then multiply those counts straight across.
Adding the groups counts the items on the shelf, not the pairs you can make.
How it works
👀 What's going on?
A canteen offers a few breads and a few fillings, and someone asks how many different sandwiches are possible. Listing them works until the numbers grow, and then it stops working.
💡 Here's the trick
Look at the tree above. Every choice in the first group joins with every choice in the second. Each new group multiplies the branches rather than adding to them.
🪄 How to do it
- 1.Count the branches at each level of the tree above.
- 2.Write those counts in a row, one per group.
- 3.Multiply them together.
✍️ Try it
A shop has 5 caps and 6 scarves; how many cap-and-scarf pairs? Then: 2 breads, 3 fillings and 2 sauces make how many sandwiches?
Each of the 5 caps pairs with each of the 6 scarves, giving 30 pairs. For the sandwich, 2 times 3 times 2 comes to 12.
⚠️ Watch out
You might be thinking you add the groups, because 5 caps and 6 scarves sounds like 11 things. Eleven is how many items sit on the shelf. The question asks for pairs, and every cap can go with every scarf.
🌟 Why this matters
Menus, outfits, passwords and timetables are all counted this way, which is why a four-digit PIN has ten thousand possibilities.
Exam tip: Draw one slot per group, write the count in each, and multiply straight across.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“3 starters, 4 mains, 2 desserts. How many 3-course meals?”
- 1
Three independent groups
Each starter pairs with each main, and each of those with each dessert.
- 2
Multiply all three
3 × 4 × 2 = 24
- 3
Watch the trap
Adding (3 + 4 + 2 = 9) is the classic mistake.
- 4
Answer ✓
= 24 meals
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap each scoop to pair it with every sauce. Fill all the rows.
PAIRS · ROUND 1 OF 3
2 scoops 🍨 × 3 sauces 🍫
Quick check
✅ VERIFICATION
A lock has 3 wheels, and each wheel shows the digits 0 to 9. How many different codes are possible?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Classification
Both, only this, only that, neither. They must add back.