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Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example,for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. If all possible orders of these $20$ people are considered,what is the average value of $S$?

(a) 9 (b) 10 (c) 11 (d) 12 (e) 13

User Aldrinleal
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1 Answer

3 votes

Answer:


$\boxed{(c) : 11}$

Explanation:

We can think of this problem as placing $7$ dividers (representing the boys) in $20$ slots (representing the people). Thus, the number of ways to arrange the boys and girls is the number of ways to place the dividers, which is $\binom{20}{7}$.

To find the average value of $S$, we need to find the total number of arrangements where $S = k$ for each possible value of $k$, then divide by the total number of arrangements.

Let's first consider the case where $k=0$. There is only $1$ way to arrange the boys and girls such that there are no boys and girls standing next to each other: BBBBBBBBBBBBBBBBBBBBB.

Next, let's consider the case where $k=1$. There are $\binom{7}{1}\binom{13}{6}$ ways to arrange the boys and girls such that there is exactly $1$ pair of boys and girls standing next to each other. To see this, consider that there are $\binom{7}{1}$ ways to choose the position of the divider that separates a boy and a girl, and $\binom{13}{6}$ ways to arrange the remaining dividers and slots.

We can continue this process to find the total number of arrangements for each possible value of $k$. When we add up all of these arrangements and divide by the total number of arrangements, we get the average value of $S$.

More specifically, the average value of $S$ is equal to

[\frac{1\cdot1+\binom{7}{1}\binom{13}{6}\cdot1+\binom{7}{2}\binom{13}{5}\cdot2+\binom{7}{3}\binom{13}{4}\cdot3+\binom{7}{4}\binom{13}{3}\cdot4+\binom{7}{5}\binom{13}{2}\cdot5+\binom{7}{6}\binom{13}{1}\cdot6+\binom{7}{7}\binom{13}{0}\cdot7}{\binom{20}{7}}.]

Simplifying this expression gives us $\boxed{(c) : 11}$.

User Joee
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