Suppose you are married and you and your partner attend a party with three other married couples. Several handshakes took place. No one shook hands with himself or herself or with their partner, and no one shook hands with the same person more than once. After all handshaking was completed, suppose you asked each person, including your partner, how many hands they had shaken. Each person gave a different answer. How many hands did your spouse shake?

There are $4$ married couples, so total persons are $8$.

You asked $7$ persons, and all gave different answers.

Maximum possible handshakes by one person are $6$ because a person cannot shake hands with himself or herself and cannot shake hands with spouse.

So the $7$ different answers must be:

$0,1,2,3,4,5,6$

Now, the person who shook $6$ hands must be paired with the person who shook $0$ hands.

Similarly,

$5$ is paired with $1$

$4$ is paired with $2$

The only remaining number is $3$.

So your spouse shook $3$ hands.

Final Answer: $ \boxed{(c)} $