Problem Solving

X is the set of all the pairs (p,q) where 1<=p<q<=N. If two distinct members of X have one constituent of the pairs in common, they are called "mates" otherwise they are called "non-mates". For example, If N=4, X={(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)} then (1,2) and (1,3) are mates (1,2), (2,3) are also mates, but (1,4),(2,3) are non-mates.

1. Find the number of non-mates that each member of X has for N=7.
a. 10
b. 36
c. 28
d. 21

OA a

2. If two members of X are mates, how many other members of X will be common mates of both these members for N=9?
a. 5
b. 7
c. 11
d. 10

OA b

Just think of the integers as different numbers that get grouped into pairs. Two groups are considered "mates" if the same number belongs to both groups and "non-mates" if the two groups have no overlap.

With that, rephrase the question.

Q1. For any given pair, how many other pairs will have no overlap? (use #s 1 through 7 only)

Suppose the given pair is (1,2). This question simply becomes how many other pairs can we make form the 5 unused numbers? In other words, how many ways are there to pick 2 from 5? This is 5!(2!3!) = 10.

Q2. Given two pairs that share a common number, how many other pairs will have that number? (use #s 1 through 9 only)

Suppose the initial mates are (1,2) and (1,3). there are only six other pairs that will be mates to both original mates: (1,4), (1,5), (1,6), (1,7), (1,8) and (1,9)