Supposing we select 5 integers at random from 1 to 40 inclusive.
How could we find the probability of none of these numbers being consecutive?
I know we have 40 choose 5 outcomes total. I'm having trouble figuring out how many ways two or more of the 5 numbers would be consecutive though.
Ok, so there's 40! ways of ordering the 5 1's and 35 0's total.
I guess we could group the consecutive 1's as follows:
5 1's [11111][000....00]
35 ways
4 1's [01111][010....00]
5*35 ways??
3 1's [11100][010...10]
This way seems really difficult, I'm not really sure where to start.