Two dice $$A$$ and $$B$$ are rolled. Let the numbers obtained on $$A$$ and $$B$$ be $$\alpha$$ and $$\beta$$ respectively. If the variance of $$\alpha - \beta$$ is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then the sum of the positive divisors of $$p$$ is equal to

Given: Two dice A and B are rolled. $$\alpha$$ and $$\beta$$ are the numbers obtained. Find the variance of $$\alpha - \beta$$.

Use properties of variance.

Since $$\alpha$$ and $$\beta$$ are independent: $$\text{Var}(\alpha - \beta) = \text{Var}(\alpha) + \text{Var}(\beta)$$

For a fair die: $$E[X] = \frac{7}{2}$$, $$E[X^2] = \frac{1+4+9+16+25+36}{6} = \frac{91}{6}$$

$$\text{Var}(X) = \frac{91}{6} - \frac{49}{4} = \frac{182 - 147}{12} = \frac{35}{12}$$

$$\text{Var}(\alpha - \beta) = \frac{35}{12} + \frac{35}{12} = \frac{35}{6}$$

So $$\frac{p}{q} = \frac{35}{6}$$, where $$\gcd(35, 6) = 1$$. Thus $$p = 35$$.

Find the sum of positive divisors of $$p = 35$$.

$$35 = 5 \times 7$$

Divisors: $$1, 5, 7, 35$$

Sum = $$1 + 5 + 7 + 35 = 48$$

The correct answer is Option C: $$48$$.