<s>[INST] EdExcel GCSE question about Hannah and the sweets: show that $n^2-n=90$ This is my reconstruction of the EdExcel GCSE question that has caused such a Twitter storm in the UK in the last 24 hours, along with its solution.

Hannah has a bag containing $n$ sweets, 6 of which are orange. She eats two sweets at random from the bag.
The probability that the two sweets Hannah eats are both orange is $\frac{1}{3}$. Show that $n^2 - n = 90$.

Possible follow-up (I don't know if this was part of the original): how many sweets were there in total in the bag?
 [/INST] Here is an alternative solution:
The probability that the first two sweets selected from a bag are orange if six of the $n$ sweets in the bag are orange is 
\begin{align*}
\frac{\binom{6}{2}}{\binom{n}{2}} & = \frac{1}{3}\\
\frac{\frac{6 \cdot 5}{2 \cdot 1}}{\frac{n(n - 1)}{2 \cdot 1}} & = \frac{1}{3}\\
\frac{6 \cdot 5}{n(n - 1)} & = \frac{1}{3}\\
\frac{30}{n(n - 1)} & = \frac{1}{3}\\
90 & = n(n - 1)\\
90 & = n^2 - n\\
\end{align*}
To determine the number of sweets in the bag, we factor the equation $n^2 - n - 90 = 0$.
\begin{align*}
n^2 - n - 90 & = 0\\
(n - 10)(n + 9) & = 0
\end{align*}
Hence, $n = 10$ or $n = -9$.  Since the number of sweets in the bag cannot be negative, the number of sweets in the bag is $10$.
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