Let n represent the number of people at the party (not including the straggler), and k represent the number of people the straggler knows.
The first part involves C(n, 2) handshakes, that is, for every pair of people there is one handshake, and there are C(n, 2) ways to pair the people.
The second part is simply k.
So C(n, 2) + k = 68.
Now we might be suspicious that the problem does not give enough information because there are two unknowns and one equation. BUT we know 0 <= k <= n. So take a look at:
C(11, 2) = 55
C(12, 2) = 66
C(13, 2) = 78
The maximum value of C(n, 2) + k for n = 11 is 66, too low.
The minimum value of C(n, 2) + k for n = 13 is 78, too high.
Therefore n = 12, and k = 2.