24. A list of 5 positive integers has mean 5 , mode 5 , median 5 and range 5 . How many such lists of 5 positive integers are there?
A 1
B 2
C 3
D 4
E 5
24. B As the mode of the list is 5 , at least two of the integers in the list are 5 . The median of the list is 5 , so when the list is written in ascending order the third integer is 5 . So, since the range of the list is 5 , the two possible lists, in ascending order, are $x, y, 5,5, x+5$ or $x, 5,5, y, x+5$, where $x$ and $y$ are positive integers and $x \leqslant y \leqslant x+5$. The mean of the integers is 5 , so their sum is $5 \times 5=25$, for each possible list. Therefore $x+y+5+5+x+5=25$. So $2 x+y=10$. The positive integer solutions of this Diophantine equation are $(1,8),(2,6),(3,4),(4,2)$. Note that neither $x$ nor $y$ equals 5 , so there are exactly two 5 s in the list. Of the solutions, neither $(1,8)$ nor $(4,2)$ satisfy the condition $x \leqslant y \leqslant x+5$.
So the only two possible lists are $2,5,5,6,7$ and $3,4,5,5,8$.