10. Alberto, Barbara, and Carlo are playing cards. In each hand, the winner gains 2 points, while the other two players each lose 1 point. Initially, everyone has 0 points. What is the probability that, after 10 hands, they are all back to zero points?
(A) 0
(B) $\frac{1}{5}$
(C) $\frac{1}{3}$
(D) $\frac{\left(\begin{array}{c}10 \\ 6\end{array}\right)}{3^{10}}$
(E) $1-\left(\frac{2}{3}\right)^{10}$.
10. The answer is (A). We want to prove that it is impossible for all players to have zero points again after 10 hands.

Let $v$ be the number of hands won by the first player when everyone ends up with 0 points, and $p$ the number of hands lost by him.

Then, according to the rule given in the text on the assignment of points, we find

$$
0=2 k-p
$$

By assumption, we have $k+p=10$, the total number of hands played; substituting in the previous equation we get $0=2 k-(10-k)$, that is, $10=3 k$.

The only solution to this equation is $k=\frac{10}{3}$, which is absurd since $k$, representing the number of hands won by the first player, should be an integer. The supposed situation is therefore impossible, and the probability that all players have 0 points is thus 0.