Math Detours · Grade 5 · Around the corner · 27

In what order can you get dressed?

Socks before shoes, but socks and trousers can go in either order. How many complete dressing orders obey every rule?

Name ____________________Date ____________

Tonight's idea

Some jobs have to happen before others. Draw an arrow from the earlier job to the later one.

For four clothes, use these rules:

You can follow arrows through the diagram. Underwear must come before shoes because there is a path through trousers, so we do not need a shortcut arrow.

There is no path from socks to trousers or from trousers to socks. These two jobs are incomparable: the rules allow either one first. They are different jobs, not equal jobs.

An ordered set can say that one thing must come before another while leaving other pairs incomparable.

Arrows record what is forced. A finished list chooses between everything.

Watch one: find every dressing order

the rules underwear trousers socks shoes all valid complete orders 1 2 3 So U T Sh U So T Sh U T So Sh U underwear · T trousers · So socks · Sh shoes
A rule diagram leaves choices open. Each row on the right makes every choice and obeys every arrow.

The shoes go last. Underwear is before trousers. Place socks in the three possible slots in the chain underwear, trousers, shoes: before underwear, between underwear and trousers, or between trousers and shoes.

Those are the three rows, so the list is complete rather than just three examples.

Your turn

  1. Add a shirt with no rules. Write “shirt” on a scrap of paper and place it into each of the five slots in every order above. How many valid orders are there now? Explain why none is counted twice.
  2. A morning has four jobs: wake, dress, eat and leave. Wake must be before dress and eat. Dress and eat must both be before leave. Draw the arrow diagram, then list every valid complete order.
  3. In the original four-clothes diagram, list every pair of jobs that is incomparable. Explain what incomparable means without using the word “equal”.
  4. Four jobs in one chain have only one valid order. Four jobs with no rules have \(4\times3\times2\times1=24\) orders. Explain each count. Why is the clothing answer, 3, between these extremes?

Stretch: add a jacket

Keep the free shirt from exercise 1. Add a jacket, with just one new rule: shirt before jacket. How many orders of all six items obey every rule?

Hint: For each original four-item order, choose two of the six positions for shirt and jacket. Once the positions are chosen, their order is forced.

Answers

  1. 15. Each of the three old orders has five slots: before all four items, in any of the three gaps, or after all four. That gives \(3\times5=15\). Removing the shirt recovers exactly one old order and one slot, so two constructions cannot give the same result.
  2. The arrows are wake to dress, wake to eat, dress to leave, and eat to leave. The valid orders are wake, dress, eat, leave and wake, eat, dress, leave. Dress and eat may swap; wake is first and leave is last.
  3. Underwear and socks; trousers and socks. Incomparable means the rules do not force either member of the pair to come first. The shoes are comparable with underwear because following arrows through trousers shows underwear before shoes.
  4. A chain forces every next job, so there is one order. With no rules, there are 4 choices for the first job, then 3, then 2, then 1: \(4\times3\times2\times1=24\). The clothing rules forbid most orders but still leave a choice about where socks go, so the answer is more than 1 and less than 24.

Stretch. 45. For one old four-item order, choose the two positions occupied by the shirt and jacket. There are \(6\times5=30\) ordered choices, but choosing the same two positions in the opposite order repeats them, so \(30\div2=15\). Shirt must use the earlier position. With three old orders, \(3\times15=45\).

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