Six in a Subset of S

To choose a subset of S, we have 2 options for each element: to be chosen / not to be chosen.

So by rule of product (multiplication principle of counting),

total Number of subsets of S = $2\cdot2\cdot2\cdot\cdot\cdot\cdot2$ (9 times) $= 2^{9} = 512$

This includes empty set (not choosing any element).

Number of subsets of S excluding empty set = $2^{9} - 1 = 511$

Let element 6 be included. No we have 2 choices (to be chosen / not to be chosen) each of remaining 8 elements.

So no. of such subsets = $2^8 = 256$

So probability = $\frac{256}{511}$

Answer: $256 + 511 = \boxed{767}$

Note: Nothing special about element 6 in this question. Any other would give same answer.