5. In the expression below, ten different digits (0 through 9) will be placed on the ten dots, where 0 cannot be at the beginning of a two-digit number:

$$
\frac{\cdots+\cdots+\cdots}{\cdots-\cdots}
$$

What is the largest possible value of this expression?
A) $\frac{255}{7}$
B) $\frac{219}{2}$
C) 116
D) 222
E) 255
A5. D) 222 The fraction $\frac{83+74+65}{20-19}$ results in 222. We can see that this is also the largest possible result as follows.

Firstly, note that the numerator is always less than $100+100+100=300$. If the denominator is greater than 1, the result is therefore less than $\frac{300}{2}=150$. To get a result of 222 or more, the denominator must therefore be equal to 1.

The only ways to get a denominator of 1 are through $20-19, 30-29, \ldots, 80-79$. In all cases, there is a 9 in the denominator, so the largest digits left for the numerator are 8, 7, 6, 5, 4, and 3. To maximize the numerator, we place the three largest digits in the tens place. The placement of the remaining three digits does not matter for the result: $80+70+60+5+4+3=222$.