Concept 4 of 7
Possible Combinations: For Every X, Every Y
Today's big idea
Every choice on one side pairs with every choice on the other, so multiply.
3 shirts against 4 trousers gives 3 x 4 = 12 different outfits.
How it works
👀 What's going on?
You own some shirts and some pairs of trousers. How many different outfits is that? Listing them all works, but it takes an age, and one multiplication does the very same job.
💡 Here's the trick
Look at the grid above. Each of the 3 shirts can be worn with any of the 4 pairs of trousers, so every shirt starts 4 outfits. Three shirts times 4 trousers gives 12 outfits altogether. Multiplying works because every choice on one side pairs with every choice on the other.
🪄 How to do it
- 1.Count the choices in the first group, as the picture above counts 3 shirts.
- 2.Count the choices in the second group.
- 3.Multiply the two counts, and that product is your answer.
✍️ Try it
A cafe has 3 juices and 5 snacks. How many juice-and-snack pairs? Then, if it adds a fourth juice, how many pairs are there?
3 juices and 5 snacks make 3 x 5 = 15 pairs. Adding one more juice makes 4 x 5 = 20. That single extra juice added 5 new pairs rather than 1, because it pairs with every snack on the list.
⚠️ Watch out
You might be thinking you add the two groups instead of multiplying. Adding 3 shirts and 4 trousers gives 7, which is how many garments you own, not how many outfits. An outfit takes one from each side, so the answer is 12.
🌟 Why this matters
How many meal deals a shop offers, or how many colour choices a bike comes in, are both worked out this way. Multiplying beats listing the moment the numbers grow.
Exam tip: Ask whether the question wants how many things or how many pairs, because that decides between adding and multiplying.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“You have 2 bikes and 4 colours. How many bike-and-colour choices?”
- 1
Two independent groups
Every bike can come in every colour.
- 2
Multiply the group sizes
2 × 4 = 8
- 3
Answer ✓
= 8 choices
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 shop sells 6 ice cream flavours in 3 types of cone. How many different cones are possible?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Position in a Row
Why counting from each end always gives one extra.