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Lining up

Counting and combinations · level 1, lesson 3 of 6

A question to start

Three friends, Ana, Ben and Cai, line up for the slide every day. They decide: "Let's stand in a different order every day, until we run out of new orders." How many days can they keep this up?

Some people guess 3, because there are 3 friends. Some guess 9. Let's find out, by acting it out and then by thinking. The answer is a number you will meet again and again in counting.

The idea

What is a line-up?

A line-up (grown-ups also say an arrangement or an order) is the friends standing one behind the other: someone is first, someone is second, someone is last. Two line-ups are different if anybody is standing in a different place.

So "Ana, Ben, Cai" and "Ben, Ana, Cai" are different line-ups: Ana is first in one and second in the other. Order matters, just like the ice-cream towers in the last lesson. A line-up is a tower lying on its side!

Act it out

If you can, find two friends (or use three toys, or three cups with letters on them) and try it. Here is a way to stay tidy, the same trick as before: keep the front person still and move only the others.

  1. Ana stands at the front. Behind her, Ben and Cai can stand two ways: Ben then Cai, or Cai then Ben.
  2. Now Ben goes to the front. Behind him, Ana and Cai can stand two ways: Ana then Cai, or Cai then Ana.
  3. Now Cai goes to the front. Behind her, Ana and Ben can stand two ways: Ana then Ben, or Ben then Ana.
Ana firstBen firstCai first ABC ACB BAC BCA CAB CBA the front of the line is on the left

Count them: 2 and 2 and 2 make 6 line-ups. The friends can stand in a new order for 6 days. On day 7 they must repeat one.

Why exactly 6?

Let's make sure we have not missed one or counted one twice.

Nothing is missed. Every line-up has somebody at the front. That somebody is Ana, Ben or Cai, so every line-up belongs to one of our three groups. Inside a group, the front person is fixed, and the other two friends can only stand in 2 ways (one of them goes second, the other goes last). We wrote both ways in every group. So every possible line-up is somewhere in the picture.

Nothing is counted twice. Two line-ups in different groups have a different person at the front, so they are different. Two line-ups in the same group have a different person second. So all 6 are different.

That is a complete reason. There are exactly 6.

Two friends, three friends, four friends

Let's look at smaller and bigger groups.

  • 1 friend: only 1 way. Ana stands there alone.
  • 2 friends: 2 ways. Ana then Ben, or Ben then Ana.
  • 3 friends: 6 ways, as we found. 3 groups (one for each front person) of 2.

Now 4 friends: Ana, Ben, Cai and Dee. Use the same trick. Pick who goes at the front: 4 choices, so 4 groups. In each group, the other 3 friends line up behind the front person, and we already know that 3 friends can line up in 6 ways. So there are 4 groups of 6:

6 and 6 and 6 and 6 make 24 line-ups.

friends 1 2 3 4
line-ups 1 2 6 24

Look how fast that grows! Each time one more friend joins, the number of groups is the number of friends, and each group holds all the line-ups of the smaller team. With 5 friends there would be 5 groups of 24, which is 120 line-ups. That is about four months of new orders, one each day.

Building on what you know

Notice what we did for 4 friends. We did not list all 24. We used the answer for 3 friends (6) as a building block. This is a very powerful habit in mathematics: solve a smaller problem first, then use it to solve the bigger one.

Worked examples

1. Three toys on a shelf. A car, a duck and a robot stand in a row. How many ways can you arrange them?

This is exactly the same as three friends lining up: the toys are the "friends", and the shelf is the line. Car first: car, duck, robot or car, robot, duck. Duck first: 2 ways. Robot first: 2 ways. 6 ways.

2. The letters of CAT. How many ways can you write the letters C, A and T in a row? (They don't all have to make real words.)

C first: CAT, CTA. A first: ACT, ATC. T first: TCA, TAC. 6 ways, and two of them (CAT and ACT) are real words.

3. Ana must go first. Ana says: "I'm always first!" Now how many line-ups of Ana, Ben and Cai are there?

Only the "Ana first" group is allowed: Ana, Ben, Cai and Ana, Cai, Ben. 2 line-ups. A rule (grown-ups say a condition) can make the count smaller.

4. Two red blocks and one blue block. You line up 3 blocks in a row: 2 red ones (exactly alike) and 1 blue one. How many different-looking rows?

If the red blocks were different, there would be 6 rows. But they look the same, so some rows look identical. The easy way: just ask where the blue block goes. It can be first, second or last:

blue, red, red · red, blue, red · red, red, blue.

3 rows. The 6 rows have paired up into 3, because swapping the two red blocks changes nothing you can see. (This is the "double scoop" idea again from the ice-cream lesson.)

Try it

  1. Two friends, Pip and Max, line up. List all the line-ups.
  2. List all 6 ways to line up a cat, a dog and a fish.
  3. Three runners, Zoe, Yan and Xavi, finish a race, and there are no ties. In how many different orders can they finish?
  4. Ana, Ben and Cai line up, but Cai wants to be last. How many line-ups are allowed? List them.
  5. How many ways can you write the letters of the word TOP in a row? How many are real English words?
  6. Four friends line up. How many line-ups have Dee at the front?
  7. You line up 3 blocks: 2 green (alike) and 1 orange. How many different-looking rows? What if there are 4 blocks: 3 green and 1 orange?
Answers
  1. 2: Pip, Max and Max, Pip.
  2. Cat first: cat, dog, fish and cat, fish, dog. Dog first: dog, cat, fish and dog, fish, cat. Fish first: fish, cat, dog and fish, dog, cat. 6.
  3. 6. A finishing order is a line-up: first, second, third.
  4. 2: Ana, Ben, Cai and Ben, Ana, Cai. With Cai fixed at the back, only Ana and Ben move, and two people can stand in 2 ways.
  5. 6 ways: TOP, TPO, OTP, OPT, PTO, POT. Real words: TOP, POT and OPT (to opt means to choose), so 3.
  6. 6. With Dee fixed at the front, the other 3 friends line up behind her, and 3 friends can line up in 6 ways. (This is one of the 4 groups of 6 that made 24.)
  7. 3 rows: the orange block is first, second or last. With 3 green and 1 orange: the orange block is first, second, third or last, so 4 rows.

Common mistake

Saying "3 friends, so 3 ways". Three ways is how many people could be at the front. But each front person starts a whole group of line-ups. Always ask: after I choose the front, how many ways are there to finish?

Missing one because you jumped around. "ABC, BCA, CAB, ACB, CBA..." Did you get them all? It is hard to tell! Keep the front person still and finish their group before moving on.

Where it leads