196201

we did the solution for button up some more. we found out that there was a pattern. we did the top button first, and there was 6 ways to button up. these were all six for number one:
1 1 1 1 1 1
2 2 3 3 4 4
3 4 4 2 3 2
4 3 2 4 2 3

then we did the second button first and there was 6 solutions. this happened very time until we did the equation 6x4=24, so there are 24 ways to button up 4 buttons. We predict that if there were 5 buttons, there would be an extra 8 ways. therefore, if there were 5 buttons, there would be 32 ways to button up, because 24+8=32.