# A game of In-between

**Discrete Mathematics**Level 5

Hayley has a deck of \(x\) cards, each labelled distinctly with an integer from 1 to \(x\), where \(x\geq 3\). She randomly picks 2 cards from the deck and sees that they have the numbers \(a\) and \(b\) written on them, where \(b>a\). Without replacing those 2 cards, she then randomly takes another card from the deck, which has the number \(c\) written on it.

Define the function \(p(x)\) to be the probability that \(a<c<b\) for a deck of \(x\) cards. Find the value of

\[\displaystyle \sum_{k=3}^{2015} {p(k)}\]

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**That seems reasonable.**Find out if you're right!

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