BEGIN BY COUNTING THE NUMBER OF RELATIONSHIPS THAT EXIST AMONG THE...
19. Begin by counting the number of relationships that exist among
the 7 individuals whom we will call A, B, C, D, E, F, and G.
First consider the relationships of individual A: AB, AC, AD, AE, AF,
AG = 6 total. Then consider the relationships of individual B without
counting the relationship AB that was already counted before: BC,
BD, BE, BF, BG = 5 total. Continuing this pattern, we can see that C
will add an additional 4 relationships, D will add an additional 3
relationships, E will add an additional 2 relationships, and F will add
1 additional relationship. Thus, there are a total of 6 + 5 + 4 + 3 +
2 + 1 = 21 total relationships between the 7 individuals.
We are told that 4 people have exactly 1 friend. This would account
for 2 "friendship" relationships (e.g. AB and CD). We are also told
that 3 people have exactly 2 friends. This would account for
another 3 "friendship" relationships (e.g. EF, EG, and FG). Thus,
there are 5 total "friendship" relationships in the group.
The probability that any 2 individuals in the group are friends is