There are 12 intermediate bus stops between two places A and B. In how many ways can a bus be made to stop at 4 of these 12 intermediate bus stops provided no two of them are consecutive?

Let us name the stops at which the train will stop as A and the stops at which it will not as B.
We need to arrange 4A's and 8B's such that no two A's come together.
_A_A_A_A_
After arranging 4 A's, we get 5 empty places which can be filled. As no two A's come together, the middle 3 spaces must have atleast 1 B.
We will name the spaces as p, q, r, s and t.
p + q + r + s + t = 8. However, q, r and s should have atleast a value of 1.
p + q1 + r1 + s1 + t = 5.
Using stars and sticks formula, the B's can be arranged in ==> (n+r-1)C(r-1) where n = 5 and r = 5.
Total ways of arrangement = 9C4 = 126 ways.