The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are chosen from the set $$\{1, 2, 3, 4, 5, 6, 7, 8\}$$. The probability of this equation having repeated roots is :

Repeated roots requires $$b^2 = 4ac$$ with $$a, b, c \in \{1,...,8\}$$. Since $$b^2$$ must be divisible by 4, $$b$$ must be even.

$$b = 2$$: $$ac = 1$$. Pairs: $$(1,1)$$. Count = 1.

$$b = 4$$: $$ac = 4$$. Pairs: $$(1,4),(2,2),(4,1)$$. Count = 3.

$$b = 6$$: $$ac = 9$$. Pairs from $$\{1,...,8\}$$: $$(3,3)$$ only. Count = 1.

$$b = 8$$: $$ac = 16$$. Pairs: $$(2,8),(4,4),(8,2)$$. Count = 3.

Total favorable = 1 + 3 + 1 + 3 = 8. Total outcomes = $$8^3 = 512$$.

Probability = $$\frac{8}{512} = \frac{1}{64}$$.

The correct answer is Option 2: $$\frac{1}{64}$$.