195993

My method that I have used is called using 'binary combinations'. I started with the top button and I have called this 1. Then I used this to work out how you can button all of them:
1,2,3,4
1,2,4,3
1,3,2,4
1,3,4,2
1,4,3,2
1,4,2,3
Since you have worked the first part out then you times by 4 to get 24 ways (6x4).