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Play: How many ways?

Counting and combinations · step 1 of 5

A question to start

You have 3 T-shirts (orange, blue and green) and 2 pairs of shorts (black and white). How many different outfits can you wear?

3 T-shirts2 pairs of shorts

Many people guess 5, because 3 and 2 make 5. But each T-shirt can go with either pair of shorts. Count them: orange with black, orange with white, blue with black, blue with white, green with black, green with white. That is 6 outfits. Counting carefully, so that you miss nothing and count nothing twice, is a real part of mathematics, called combinatorics.

The idea

Make an organised list

The secret of counting is to be tidy. If you write outfits down in any old order, you will lose track. Instead, pick the first thing and go through every choice for the second thing. Then move on to the next first thing.

  • Orange T-shirt: with black, with white.
  • Blue T-shirt: with black, with white.
  • Green T-shirt: with black, with white.

Each T-shirt gets its own line, and every line has the same 2 shorts. Three lines of 2 is 6.

Draw a tree

A tree diagram shows the same thing as a picture. Start at one point. Draw a branch for each first choice. From the end of each branch, draw a branch for each second choice. Every path from the start to a tip is one outfit, so you just count the tips.

blackwhite blackwhite blackwhite 6 tips

Handshakes at a party

Four friends, Ana, Ben, Cai and Dee, meet at a party. Everyone shakes hands with everyone else, once. How many handshakes are there?

Draw four dots for the friends and a line for each handshake.

AnaBenCaiDee

There are 6 lines, so 6 handshakes. You can also count them tidily:

  • Ana shakes hands with Ben, Cai and Dee: 3 handshakes.
  • Ben has already shaken Ana's hand. He still shakes Cai and Dee: 2 more.
  • Cai has already met Ana and Ben. She still shakes Dee: 1 more.
  • Dee has already shaken everyone's hand: 0 more.

3 + 2 + 1 = 6. The tidy list never counts "Ana and Ben" and "Ben and Ana" twice, because they are the same handshake.

When the order matters

Now three toys, a car, a duck and a robot, stand in a row on a shelf. How many different ways can you line them up? This time the order does matter: car, duck, robot looks different from robot, duck, car.

Choose the first toy, then go through the rest:

  • Car first: car, duck, robot or car, robot, duck.
  • Duck first: duck, car, robot or duck, robot, car.
  • Robot first: robot, car, duck or robot, duck, car.

6 ways. Three choices for the first place, and then 2 ways to finish each one.

Worked example

An ice-cream shop sells three flavours: vanilla, strawberry and chocolate. You may choose two different flavours in a cup. How many different cups can you have?

  1. Make a tidy list. Start with vanilla and pair it with each of the others: vanilla and strawberry, vanilla and chocolate.
  2. Now strawberry. It has already been paired with vanilla, so only one new cup: strawberry and chocolate.
  3. Now chocolate. It has already been paired with both the others. Nothing new.
  4. Count: 2 + 1 = 3 cups.

Now change the question: you want two scoops in a cone, one on top of the other, and you care which is on top. Then vanilla on top of strawberry is different from strawberry on top of vanilla, so every cup turns into 2 cones: 3 × 2 = 6 cones. Whether the order matters changes the answer. You will see this idea again and again in the steps that follow.

Try it

  1. You have 2 hats and 3 scarves. How many hat-and-scarf outfits can you make?
  2. Five friends meet and everyone shakes hands with everyone else once. How many handshakes?
  3. In how many ways can you arrange the letters of the word CAT? List them.
  4. Using the digits 1, 2 and 3, how many two-digit numbers can you make if you may not use a digit twice?
  5. A pizza shop has 3 toppings: cheese, mushroom and pepper. You can have any of them on your pizza, or none at all. How many different pizzas are there?
Answers
  1. 6. Each hat goes with any of the 3 scarves: 3 outfits for the first hat and 3 for the second.
  2. 10. The first friend shakes 4 hands, the next 3 new ones, then 2, then 1, then 0: 4 + 3 + 2 + 1 = 10. Drawing 5 dots and joining every pair also gives 10 lines.
  3. 6: CAT, CTA, ACT, ATC, TCA, TAC. Three choices of first letter, and 2 ways to finish each.
  4. 6: 12, 13, 21, 23, 31, 32. Order matters here: 12 and 21 are different numbers.
  5. 8. For each topping you decide "yes" or "no". List them tidily: none; cheese; mushroom; pepper; cheese and mushroom; cheese and pepper; mushroom and pepper; all three. A tree with a yes/no split for each topping has 2, then 4, then 8 tips.

Common mistake

Counting the same thing twice. When Ana shakes Ben's hand, Ben also shakes Ana's hand, but it is one handshake. If you count "4 friends each shake 3 hands, so 12", you have counted every handshake twice: the true answer is 12 ÷ 2 = 6. Always ask: does swapping the order give something new, or the same thing?

Where it leads