[Solution] Since there are 7 possible scenarios for drawing the balls such that the sum of the numbers on the balls is 6, and they are equally likely.
We can represent these scenarios with the following ordered triples:
$$
(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) \text { and }(2,2,2) \text {, }
$$

Therefore, the probability that the ball marked 2 is drawn all three times is $\frac{1}{7}$.
Hence, the answer is (C).
$14 \cdot 21$ Three balls marked $1, 2, 3$ are placed in a jar. A ball is drawn from the jar, its number is recorded, and then it is returned to the jar. This process is repeated three times. Each ball has an equal chance of being drawn each time. If the sum of the recorded numbers is 6, what is the probability that the ball marked 2 was drawn all three times?
(A) $\frac{1}{27}$.
(B) $\frac{1}{8}$.
(C) $\frac{1}{7}$.
(D) $\frac{1}{6}$
(E) $\frac{1}{3}$.
(34th American High School Mathematics Examination, 1983)