1. Identify the distribution

There are:

• $3$ rotten apples
• $7$ good apples
• total $10$ apples

We draw $4$ apples without replacement.

If $X$ = number of rotten apples drawn, then $X$ follows a hypergeometric distribution with parameters: $N=10,\quad K=3,\quad n=4$ where:

• $N$ = total population size
• $K$ = number of rotten apples
• $n$ = number of draws

2. Find the mean $\mu$

For a hypergeometric random variable, $\mu = n\cdot \frac{K}{N}$ So, $\mu = 4\cdot \frac{3}{10} = \frac{12}{10} = \frac{6}{5}$

Thus, $\mu = \frac{6}{5}$

3. Find the variance $\sigma^2$

For a hypergeometric random variable, $\sigma^2 = n\cdot \frac{K}{N}\left(1-\frac{K}{N}\right)\cdot \frac{N-n}{N-1}$ Substitute the values: $\sigma^2 = 4\cdot \frac{3}{10}\cdot \frac{7}{10}\cdot \frac{10-4}{10-1}$ $= 4\cdot \frac{3}{10}\cdot \frac{7}{10}\cdot \frac{6}{9}$ Now simplify: $4\cdot \frac{3}{10} = \frac{12}{10} = \frac{6}{5}$ So, $\sigma^2 = \frac{6}{5}\cdot \frac{7}{10}\cdot \frac{2}{3}$ $= \frac{42}{50}\cdot \frac{2}{3} = \frac{21}{25}\cdot \frac{2}{3} = \frac{42}{75} = \frac{14}{25}$

Thus, $\sigma^2 = \frac{14}{25}$

4. Compute $\mu^2 + \sigma^2$

$\mu^2 = \left(\frac{6}{5}\right)^2 = \frac{36}{25}$ Therefore, $\mu^2 + \sigma^2 = \frac{36}{25} + \frac{14}{25} = \frac{50}{25} = 2$

5. Compute the required value

$10(\mu^2+\sigma^2)=10\times 2=20$

6. Check options

The correct option is: $\boxed{20}$ So, Option A is correct.