My wife did not shake hands with anyone at the party.
Let's start by assuming that each person shook hands with a different number of people. If everyone shook hands with a different number of people, then there must have been a total of 10 handshakes (since there were 10 people at the party).
Now, let's consider how many hands each person could have shaken. Since no one shook hands with themselves or their spouse, and since each person shook hands with a different number of people, the number of hands each person shook must be one of the following:
0, 1, 2, 3, or 4.
Since there were 10 handshakes in total, and since no one shook hands with the same person more than once, we can conclude that the sum of the number of hands shaken by each person must be 10.
Let's assume that the number of hands shaken by each person, in increasing order, is a, b, c, d, e, f, g, h, i, j. Then we have:
a + b + c + d + e + f + g + h + i + j = 10
Since each person gave a different answer, we can conclude that each of the numbers a, b, c, d, e, f, g, h, i, j must be different.
Now, let's consider which combinations of numbers from the set {0, 1, 2, 3, 4} can be used to add up to 10. We can start with the largest number, 4, and see that it can only be used once, since no one shook hands with the same person more than once. Therefore, the only way to get a sum of 10 is:
4 + 3 + 2 + 1 = 10
Now, let's assign these numbers to the people in increasing order, starting with my wife. Let's say that my wife shook hands with "x" people. Then we have:
My wife: x
Person 2: 4
Person 3: 3
Person 4: 2
Person 5: 1
Person 6: 0
Person 7: 0
Person 8: 0
Person 9: 0
Person 10: 0
Since no one shook hands with their spouse, my wife must have shaken hands with each of the other four married couples, for a total of 4 people. Therefore, we have:
x + 4 + 3 + 2 + 1 + 0 + 0 + 0 + 0 + 0 = 10
Simplifying this equation, we get:
x = 0
Therefore, my wife did not shake hands with anyone at the party.