Answer: Option A
Let O denote the set with guests who have shook odd number of hands and E denote the set with guests who have shook even number of hands at any point of time in the party. Suppose two guests G1 and G2 have just shook hands. We have the following 2 cases
Case1: G1 and G2 are from same set. Suppose they both were in set O before handshake. After handshake, they are both now in set E. |O| = |O|(old) – 2 and |E| = |E|(old) + 2. Same logic for set E.
Case 2: G1 and G2 are from different sets. After shaking hands, they exchange their sets. Suppose G1 is in O and G2 is in E. After handshake G1 is in E and G2 is in O and vice versa. Size of both sets remain unchanged.
In any case, size of a set changes by 2 or remains unchanged. Thus Option A is the answer.