Hanger and Cupboard Probabilty question!

Question

Your cupboard has 10 hangers arranged in a row to hang clothes. There are 7 clothes hung.

Assume you picked hangers at random among the spaces available. Find the probability that

the three empty hangers are adjacent to each other.

Answer 1

There are 8 places the 3 empty hangers could be: 1-3, 2-4, 3-5, ..., 8-10

There are C(10,3) places where we could have the empty hangers.

Thus, the probability that the empty hangers are together is

8/C(10,3) = 8/120 = 1/15