How many man-woman neighbors in a random line?

Question

Five men and five women (10 people total) stand in a single row in a completely random order. On average, how many of the 9 neighboring pairs consist of one man and one woman?

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A row of 10 has 9 neighboring pairs. Put one indicator on each pair and add. Fix one neighboring pair of positions. The two people there are a random draw of 2 from the 10 without replacement. A mixed pair happens as "man then woman" or "woman then man": P(	ext{mixed}) = \frac{5}{10}\cdot\frac{5}{9} + \frac{5}{10}\cdot\frac{5}{9} = 2 \cdot \frac{25}{90} = \frac{50}{90} = \frac59. Let Xi = 1 if neighboring pair i is mixed; E[Xi] = 	frac59. With 9 pairs, E[X] = 9 	imes \frac59 = 5. Takeaway. Each of the 9 gaps is a man-woman pair with probability 	frac59, and summing the indicators gives 9 …

How many man-woman neighbors in a random line?

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