17. Matthew rolled a normal die 24 times. All numbers from 1 to 6 came up at least once. The number 1 came up more times than any other number. Matthew added up all the numbers. The total he obtained was the largest one possible. What total did he obtain?
A 83
B 84
C 89
D 90
E 100
SolUtion
D

Since Matthew has rolled each number at least once, we can guarantee that 6 of the rolls have a sum of $1+2+3+4+5+6=21$. Let's consider the remaining 18 rolls.
There should be more $1 \mathrm{~s}$ than any other numbers. It is clear that the number of sixes rolled should be one less than the number of ones in order to make the sum as large as possible.
If we have 9 ones: then 8 sixes adding a five we get $9+48+5=62$.
If we have 8 ones: then 7 sixes adding 3 fives we get $8+42+15=65$.
If we have 7 ones: then 6 sixes adding 5 fives we get $7+36+25=68$.
If we have 6 ones: then 5 sixes, 5 fives and adding 2 fours we get $6+30+25+8$, which is 69 .
This leads to a maximum total of 90 .
Further reduction of $1 \mathrm{~s}$ does not increase the maximum possible sum.