Okay so my brother just came home and has helped me figure this out!
So 91 is the total sample space of getting at least one 6. To get 91 he said:
Imagine the first dice is a 6, the total number of options of getting a 6 in the other two dices is 6x6 = 36
Now, imagine the first dice isn't a 6, we then have the options: 61, 62, 63, 64, 65, 66, 16, 26, 36, 46, 56 - these are 11 different combinations of ways of getting at least one 6 with other other two dice.
We times this 11 by 5 as if the first dice isn't a 6, there are 5 options it can take. 55+36 makes a sample space of 91.
He started off with getting exactly two 6s
To find the sample space of getting two or more sixes -
If the first dice is a 6 and we need one or more 6s
If the second dice is a 6 we have 6 available options for the third dice
OR if the second dice still isn't a 6, in this case we have 1 available option for the third dice, we times this by 5 again as there are 5 different ways the second dice isn't a 6
Now, if the first dice isn't a six. We only have one option, and that is for the two of dices to be 6s. Again there are 5 different ways of the first dice not being a six, so 5x1 = 5
11+5 = 16 and this is our sample space of the total number of outcomes being two or more 6s.
We minus one from this as we are looking for EXACTLY two 6s, and the only option that isn't exactly two 6s are the dice 666, which appears once, this give us 15/91 for the second part.
Back to the first part, to get exactly one 6, this is the number of combinations of getting 1 or more 6s minus that of getting 2 or more 6s, this gives 91-16, which is 75 and the answer 75/91.
To be honest, I understand his workings out but if this was to pop up in an exam I'd have no clue what was going on. :/ Have completely given up on it now - really hard question!