I have typed Marcus and Jakob's explanation in case the photograph is not clear to read. However, I really wanted to include an image of their careful, systematic recording, which they worked so hard on!
They started this in school and were close to a solution, then went home and carried on thinking and came back the next day with a clear formula to solve the problem. They spent their break times writing up their thinking as they were so desperate for me to submit a solution on their behalf and are rightly proud of themselves! This is the same case for all of the children who have submitted a solution from Milton Road Primary School - I have been blown away by their perseverance and inspired by their curiosity! (From their teacher, Mrs Baucher-Webb)
They said:
To figure out the number of possibilities, you need to multiply the number of buttons by one less and less up to 1. E.g. 5 x 4 x 3 x 2 x 1 = 120
This is because the first number has 5 different spaces to go in. When you have placed it, the second number will only have 4 possibilities. Then the third number will only have 3. Then 2, then 1.
That shows that for 3 buttons (3 x 2 x 1 = 6) there are 6 different possibiliites.
4 buttons (4 x 3 x 2 x 1 = 24) has 24 different possibilities.
5 buttons (5 x 4 x 3 x 2 x 1 = 120) has 120 different possibilities.
6 buttons (6 x 5 x 4 x 3 x 2 x 1 = 720) has 720 different possibilities.
First, we did all of this recording but then we realised we didn't need to do it because of the formula above. It was a good way to double check our idea though!