The Smiths, the Andrings and the Cliffords all hold a big party. Everyone shakes hands with every member of the other two families (no one shakes hands with members of their own family), 142 handshakes in all.

Assuming that there at least as many Andrings as Smiths, and at least as many Cliffords as Andrings, determine:

A) Is there any number larger than 142 handshakes that still leads to a unique solution?

(B) What is the largest number of handshakes if we are told that there is a different number of people in each family?

(C) What is the largest number leading to a unique solution if we know that there are at least 2 people in each family?

(D) What is the maximum if there are 4 families, each with at least 2 members, and each with a different number of members?