# Number of Ways to Arrange Two Couples and A Single Person

Two couples and a single person are seated at random in a row of five chairs. What is the probability at least one person is not beside his/her partner?

Let $$P(A)$$ denote the probability that both couples are seated together in the arrangement. Therefore, the probability that at least one person is not beside his/her partner should be
$$1-P(A)$$ The total number of arrangements is $$5!$$ and the number of arrangements for both couples sitting together is $$3!2!2!$$ So, the probability that at least one per son is not beside his/her partner is $$1-\frac{3!2!2!}{5!}=\frac{4}{5}$$

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