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There are n married couples at a party. ...

There are n married couples at a party. Each person shakes hand with every person other than their spouse. The total number of hand-shakes must be:

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Total number of possible hand-shakes
=number of ways in which two persons can be selected
`= .^(2n)C_(2)`
This will include n hand-shakes of a person with her or his spouse.
`therefore` Required number of hand-shakes `= .^(2n)C_(2)-n`
=2n(n-1)
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