Handshakes pass germs. Fist bumps are better. But a "Namaste!" is the best!

N people came into a party to have some fun. It is known that some couples attended the party, but the number of people who came by themselves is odd. Anyone can give a handshake to whomever they want- it is not compulsory for anyone. Assuming that either people went to the party by themselves or along with their spouse/girlfriend/boyfriend, what is the probability that the number of people who gave an odd number of handshakes is even?

Details and Assumptions:

  • No one entered the party in groups that were larger than 2.
  • Couples who entered can give a handshake to whomever they came with- it is also counted.
  • No one gives more than one handshake to the same person once they enter the party.
  • And no one can give a handshake to themselves. That would just be weird to look at.
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