Hannah's sweet problem (Edexcel 2015): There are n sweets, 6 are orange, rest of the sweets are yellow. She takes 2 sweets randomly without replacing them and the probability that 2 orange sweets are chosen is 1/3. Show that n^2-n-90 = 0.

Draw a probability tree diagram. For this question, only two branches are required (orange and orange). At the start, there are n sweets in total, 6 are orange, so p(O) = 6/n. On our second pick, there are 5 orange sweets and the total number of sweets is n-1, so p(O) = 5/(n-1). Using the information given in the question that p(O and O) = 1/3, and using the probablities from the tree, we will arrive at the equation that we have been asked to shown. Since p(A and B) = p(A) x p(B), therefore (6/n) x (5/(n-1)) = 1/3. Manipulate this and the equation will come out.