Exercise $6\mathrm{.}3.[10$pts$]$ There are n people who can shake hands with one another $($where n $>1).$ Use pigeonhole principle to show that there is always a pair of people who will shake hands with the same number of people.Hint. Pigeonhole principle does not immediately apply to this problem. Solve the problem for two cases:$(1)$ There is no person who shakes hands with everyone else $($all handshake numbers are strictly less than n $-1).$ Easy case.$(2)$ There is a person who shakes hands with everyone else $($handshake number is n $-1).$ What can you say about handshake numbers for everyone else?