Question: > 
> If $a < x < b$, then there is one and only one number $\lambda$ with $0 < \lambda < 1$ and $x = (1-\lambda)a + \lambda b$. 
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The book calls for proof of this statement.

Answer: 
Since there are four options for each choice, and each choice is independent of the others, and there are two choices, there are $4^2=16$ total options.
You take one and you have got 3 options left. So three possibilities if you pick a first, another three if you you pick b first .... . So in total you have got 12 possibilities.