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$n$ couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

  1. \(^{2n}\mathrm{C}_n\times 2^n\)
  2. \(3^n\)
  3. \(\frac{(2n)!}{2^n}\)
  4. \(^{2n}\mathrm{C}_n\)
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We can also think in terms of boolean inputs.

Husband            Wife              Condition

0                         0                   both don’t come                                                 👍

0                         1                   wife without husband                                        👍

1                         0                   husband cannot come by himself                   👎

1                         1                   both come together                                           👍

Thus 3 valid cases!

Answer is 3^n.
Answer:

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