Each of n boys and n girls, independently and randomly, chooses a member of the other sex....

Each of n boys and n girls, independently and randomly, chooses a member of the other sex. If a boy and girl choose each other, they become a couple. Number the girls, and let Gi be the event that girl number i is part of a couple. Let  be the probability that no couples are formed,

(a) What is P(Gi)?

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(b) What is P(Gi\Gj)?

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(c) When n is large, approximate P0.

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(d) When n is large, approximate Pk, the probability that exactly k couples are formed.

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(e) Use the inclusion-exclusion identity to evaluate P0.