10. (40 points) Given positive integers $a$, $b$, and $c$ to 甲, 乙, and 丙 respectively, each person only knows their own number. They are told that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$, and each is asked the following two questions:
(1) Do you know the value of $a+b+c$?
(2) Do you know the values of $a$, $b$, and $c$ respectively?
甲 answers "no" to both questions, and 乙, after hearing 甲's answers, answers "yes" to the first question and "no" to the second question. After hearing both of their answers, how should 丙 answer these two questions?
10. First solve the equation $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$.

Assume $a \geqslant b \geqslant c$. If $c=3$, then $1=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leqslant \frac{3}{c}$.
Thus, $a=b=c=3$. Otherwise, $c=2$.
Then, $\frac{1}{a}+\frac{1}{b}=\frac{1}{2}$.
Solving this, we get $a=b=4$ or $b=3, a=6$.
The sums corresponding to these three solutions are
$$
3+3+3=9,4+4+2=10,6+3+2=11 \text {. }
$$

Since Jia's answer to question (1) is negative, therefore, $a \neq 6$ or 4.
Similarly, $b=6$ or 4.
If $b=4$, since $a \neq 4$, then $a=2, c=4$.
Since Yi's answer to question (2) is negative, hence $b=6$.
Therefore, Bing knows $b=6$ and $a+b+c=11$.
Since Bing also knows the value of $c$, he must also know the value of $a$.
Thus, Bing answers "know" to both questions.