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