Two distinct integers are selected at random from the list of integers

Between 10 and 60 inclusive we have:

- five multiples of 9
- nine multiples of 6

We can't just add these numbers (five plus nine) to work out how many numbers are multiples of six or nine, because some numbers are in both groups: if a number is a multiple of 18 (the LCM of 9 and 6) it is divisible by both. If we just add 5+9, we'll be counting every multiple of 18 twice. We have three multiples of 18 between 10 and 60, so we must subtract three to avoid counting those multiples of 18 twice, and we thus have 5+9-3 = 11 distinct numbers between 10 and 60 inclusive which are divisible by six or by nine (or by both).

So, when we pick our first number, 11 numbers 'work' out of the 51 we are choosing from. Once we pick that number, we have 50 numbers left in total, and 10 of them 'work'. So the probability both numbers 'work' is:

(11/51)(10/50) = (11/51)(1/5) = 11/255