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# 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?

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## Hints

• Hint 1
What can you say about the ordering (not necessarily the rank) of $$B,C,D?$$
• Hint 2
You deduced an ordering of $$B,C,D.$$ In how many positions can you place $$A?$$
• Hint 3
What 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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