$10$ coins weigh $10$ grams each, and another $10$ coins weigh $11$ grams each. So the average weight is:
$$\frac{10 \cdot 10 + 10 \cdot 11}{20} = 10.5\text{ grams}.$$
Now I need to distribute coins to $3$ persons: $5$ coins to A, $5$ to B and $10$ to C. How to distribute them fairly so the average weight of coins for each person is as close as $10.5$ grams?
Hint:
C can have an average of $10.5$g while A and B cannot: they can only achieve $10.4$g or $10.6$g.
You might find it easier to see this by multiplying these by the numbers of coins: C could get a total of $105$g from ten coins while A and B can only get $52$g or $53$g from five coins each.