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Exactly $2$ name matches

ProbabilityHard

$5$ traders each write their name on a chit and the chits are shuffled, then dealt one per trader, with all $5!$ assignments equally likely. What is the probability that exactly $2$ traders receive their own chit?

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Hint

Choose which traders match, then count the ways the rest all miss.

Solution

Choose the $2$ traders who get their own chit: $10$ ways.
The other $3$ traders must all receive a wrong chit; the number of such arrangements of $3$ items is $2$ .
Favorable assignments: $10 \times 2 = 20$ .
Total assignments: $5! = 120$ .
The probability is $\frac{20}{120} = \frac{1}{6}$ , so the answer is $\frac{1}{6}$ .

Answer: $\frac{1}{6}$

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