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CombinatoricsEasy

$12$ market makers each have a distinct favorite product out of $12$ products. The products are assigned one per market maker uniformly at random over all $12!$ assignments. What is the expected number of market makers who receive their own favorite product?

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Hint

Write the count as a sum of indicators and use linearity of expectation.

Solution

For each market maker, the probability of receiving their favorite product is $\frac{1}{12}$ .
By linearity of expectation, the expected count is the sum over all $12$ market makers.
This gives $12 \times \frac{1}{12} = 1$ .
The answer is $1$ .

Answer: $1$

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