# Button-Up

## Problem

My coat has three different buttons.

Sometimes, I do them up starting with the top button. Sometimes, I start somewhere else.

How many ways can you find to do up my coat?

How will you remember them?

Do you think there are any more? How do you know?

This problem was inspired by an idea of Bernard Murphy.

## Getting Started

Do you always have to start in the same place?

How will you remember the ways you've found?

## Student Solutions

You found several different ways to help find the solution to this problem.

Lydia and Megan from Moorfield drew pictures of the buttons and numbered each button according to the order it was done up. They found six ways:

Some of you described the buttons as 'top', 'middle' and 'bottom' then made a list of all the possible ways of doing them up. For example, Abbie from Oakthorpe Primary said:

top bottom middle

If we start with the middle button:

middle bottom top

middle top bottom

If we start with the bottom button:

bottom top middle

bottom middle top

Then there were those of you who labelled your buttons as $1$, $2$ and $3$, like Yousef at Levendale Primary who wrote:

Karnan from Stag Lane Junior School explained how he knew he had all the possibilities:

These are all the combinations for the buttons. You can be sure because all you have to do is:1. See how many combinations there are for buttoning the top button on first.

2. Then, you have to multiply by three for three possible starting positions.

Well done all of you. Kurtis from Moorfield School and Demi from Tudhoe Grange rightly pointed out that we were presuming we wanted to do up all three buttons. Kurtis asks:

Perhaps you could find out how many ways there are if you were allowed to do up $1$, $2$ or $3$ buttons?What a great question, Kurtis!