# Practice Paper 2 Question 2

Five candidates took a maths test and got scores \(A, B, C, D, E,\) with \(A>B,C>D,D>B,E>B\). In how many possible ways could the candidates be ranked?

## Related topics

## Hints

- Hint 1What can you say about the ordering (not necessarily the rank) of \(B,C,D?\)
- Hint 2You deduced an ordering of \(B,C,D.\) In how many positions can you place \(A?\)
- Hint 3What about the number of positions for \(E\) with respect to \(A,B,C,D?\)

## Solution

From the given inequalities, it can be deduced that \(C>D>B.\) As \(A>B,\) this leaves \(3\) positions that \(A\) can be placed in. Since \(E>B\) then regardless of the position chosen for \(A\) there are \(4\) possible positions for \(E.\) Therefore, there are \(3 \cdot 4=12\) possible ways.

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