The mean of the numbers $$a, b, 8, 5, 10$$ is $$6$$ and their variance is $$6.8$$. If $$M$$ is the mean deviation of the numbers about the mean, then $$25M$$ is equal to

We need to find $$25M$$ where $$M$$ is the mean deviation about the mean of the numbers $$a, b, 8, 5, 10$$ with mean 6 and variance 6.8.

Using the mean condition, we have $$\frac{a + b + 8 + 5 + 10}{5} = 6$$, which simplifies to $$a + b + 23 = 30$$ and hence $$a + b = 7 \quad \cdots (1)$$.

Now applying the variance condition $$\text{Variance} = \frac{\sum(x_i - \bar{x})^2}{5} = 6.8$$ leads to $$(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2 = 34$$. Substituting the known values gives $$(a-6)^2 + (b-6)^2 + 4 + 1 + 16 = 34$$, so that $$(a-6)^2 + (b-6)^2 = 13 \quad \cdots (2)$$.

Since from (1) $$b = 7 - a$$, substituting into (2) yields $$(a-6)^2 + (1-a)^2 = 13$$. Expanding leads to $$a^2 - 12a + 36 + a^2 - 2a + 1 = 13$$, which simplifies to $$2a^2 - 14a + 24 = 0$$ and then to $$a^2 - 7a + 12 = 0$$. Factoring gives $$(a-3)(a-4) = 0$$, so that $$\{a, b\} = \{3,4\}$$.

Next, the mean deviation is calculated as $$M = \frac{|3-6| + |4-6| + |8-6| + |5-6| + |10-6|}{5}$$, which is $$M = \frac{3 + 2 + 2 + 1 + 4}{5} = \frac{12}{5}$$.

Therefore, $$25M = 25 \times \frac{12}{5} = 60$$.

The correct answer is Option A.