3) What is the maximum value that the number $a(b+c)-b(a+c)$ can take when $a$, $b$, and $c$ are distinct integers, each greater than or equal to 1 and less than or equal to $10$?
(A) 80
(B) 81
(C) 84
(D) 90
(E) 100.
3. The answer is (B). $a(b+c)-b(a+c)=c(a-b)$. To make this number as large as possible, we choose $b$ as small as possible, that is, $b=1$. To maximize $c(a-1)$, we must choose $a$ and $c$ as large as possible, while ensuring they are less than or equal to 10 and distinct. For $a=10$ and $c=9$, we get 81, while for $a=9$ and $c=10$, we get 80. Therefore, 81 is the maximum value we can obtain.