A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that $$x \in A$$ is:

The set S has 7 elements. We need to find the probability that a randomly chosen non-empty subset A of S and a randomly chosen element x of S satisfy the condition that x is in A.

First, we determine the total number of possible outcomes. The total number of subsets of S is $$2^7 = 128$$, which includes the empty set. Since we are choosing a non-empty subset, we exclude the empty set. Therefore, the number of non-empty subsets is $$128 - 1 = 127$$.

For each non-empty subset A, we can choose any element x from S. Since S has 7 elements, there are 7 choices for x. Therefore, the total number of ways to choose a non-empty subset A and an element x is $$127 \times 7 = 889$$.

Next, we find the number of favorable outcomes where x is in A. We fix an element x. The number of non-empty subsets that contain x is calculated by noting that if x is included, we can choose any subset of the remaining 6 elements. The number of subsets of 6 elements is $$2^6 = 64$$. Since each of these subsets includes x, they are all non-empty. Thus, for each fixed x, there are 64 non-empty subsets containing x.

Since there are 7 possible choices for x, the total number of favorable pairs (A, x) is $$7 \times 64 = 448$$.

The probability is the ratio of the number of favorable outcomes to the total number of outcomes, which is $$\frac{448}{889}$$. Simplifying this fraction, we divide both the numerator and denominator by 7: $$448 \div 7 = 64$$ and $$889 \div 7 = 127$$, so the probability is $$\frac{64}{127}$$.

Alternatively, we can verify by fixing x and computing the probability that A contains x. For a fixed x, the number of non-empty subsets containing x is 64, and the total number of non-empty subsets is 127. Thus, the probability is $$\frac{64}{127}$$, which is the same for any x due to symmetry.

Comparing with the options, $$\frac{64}{127}$$ corresponds to Option B.

Hence, the correct answer is Option B.