To find the expected value of $S$, the number of places where a boy and a girl are standing next to each other in a line of 7 boys and 13 girls, we start by understanding the problem setup.

There are 20 positions in total, and we are interested in the number of adjacent pairs (i.e., consecutive pairs of positions) where one position is occupied by a boy and the other by a girl. There are $n-1$ pairs of adjacent positions in a row of $n$ positions. For $n = 20$, there are $19$ adjacent pairs.

Next, consider the arrangement and the expected value. Let's denote the sequence of children as follows:

• $X_i$ is an indicator variable which is 1 if the $i$-th and $(i+1)$-th positions form a boy-girl pair or a girl-boy pair, and 0 otherwise.
• $S = \sum_{i=1}^{19} X_i$

To find the expected value of $S$, we use the linearity of expectation: [ \mathbb{E}[S] = \mathbb{E}\left[\sum_{i=1}^{19} X_i\right] = \sum_{i=1}^{19} \mathbb{E}[X_i] ]

Each $X_i$ is 1 if a boy and a girl (in either order) are next to each other at positions $i$ and $i+1$.

The probability that the $i$-th position is occupied by a boy is $\frac{7}{20}$, and the probability that it is occupied by a girl is $\frac{13}{20}$.

Thus, the probability that positions $i$ and $i+1$ are a boy-girl pair is: $\mathbb{P}(B_iG_{i+1}) = \left(\frac{7}{20}\right) \left(\frac{13}{19}\right)$ And similarly, the probability that positions $i$ and $i+1$ are a girl-boy pair is: $\mathbb{P}(G_iB_{i+1}) = \left(\frac{13}{20}\right) \left(\frac{7}{19}\right)$

Summing these two probabilities, we get: $\mathbb{P}(X_i = 1) = \left(\frac{7}{20}\right) \left(\frac{13}{19}\right) + \left(\frac{13}{20}\right) \left(\frac{7}{19}\right)$ $\mathbb{P}(X_i = 1) = \frac{91}{380} + \frac{91}{380} = \frac{182}{380} = \frac{91}{190}$

The expected value of $X_i$ is: $\mathbb{E}[X_i] = \mathbb{P}(X_i = 1) = \frac{91}{190}$

Since there are 19 such pairs, the expected value of $S$ is: $\mathbb{E}[S] = 19 \times \frac{91}{190} = \frac{1729}{190} \approx 9.1$

Rounding to the nearest integer, the expected value of $S$ is: $\boxed{9}$

Would you like more details on any part of this solution, or do you have any other questions?

Here are five related questions you might find interesting:

1. What is the probability that all boys are standing next to each other?
2. How would the expected value of $S$ change if there were 10 boys and 10 girls?
3. What is the expected number of boy-boy pairs in the row?
4. If we know the exact sequence, how can we calculate $S$ directly?
5. How would the expected value of $S$ change if the positions were arranged in a circle instead of a row?

Tip: When calculating expected values, breaking down the problem into smaller components and using the linearity of expectation can simplify the process significantly.