Concept 5 of 7
Possible Combinations: For Every X, Every Y
Today's big idea
Every item in the first group pairs with every item in the second, so multiply.
Adding the two counts is the usual slip, and it always gives far too few.
How it works
👀 What's going on?
Three shirts and four pairs of trousers are laid out on the bed. Working out how many different outfits that makes feels like it should need counting, and it does not.
💡 Here's the trick
Look at the grid above. Every shirt can be worn with every pair of trousers, so each shirt opens up four outfits of its own. Three shirts each offering four outfits gives twelve, and that is a multiplication rather than a count.
🪄 How to do it
- 1.Find the two groups in the grid above and count how many are in each.
- 2.Check that any item from one group can go with any item from the other.
- 3.Multiply the two counts together.
✍️ Try it
You have 4 caps and 5 T-shirts. How many cap-and-T-shirt outfits? Then: a stall offers 3 breads and 2 fillings, and you take one of each.
Four caps times five T-shirts gives 20 outfits. Three breads times two fillings gives 6 sandwiches.
⚠️ Watch out
You might be thinking you add the two numbers. Three shirts and four trousers would then give seven, but seven is just the number of garments on the bed. The question asks for pairings, and every shirt pairs with every trouser.
🌟 Why this matters
Working out how many choices you really have, from a menu or a wardrobe, is this exact sum. It also shows why a small extra choice adds far more options than it looks like it should.
Exam tip: Ask yourself whether the answer should be bigger than either group; if it is not, you have added instead of multiplied.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“You have 3 shirts and 4 pants. How many different outfits?”
- 1
Understand the rule
Every shirt can pair with every pant. Two independent choices.
- 2
Multiply the group sizes
3 × 4 = 12
- 3
Check by tree (optional)
Shirt 1 → {P1, P2, P3, P4}. Shirt 2 → 4 pants. Shirt 3 → 4 pants. Total = 4+4+4 = 12. ✓
- 4
Answer ✓
= 12 outfits
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
You can make 12 different pairs from some shirts and 3 caps. How many shirts are there?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Who Has What
With two people and two things, one no answers everything.