My roommate and I are having a house party. We invite 15 couples. At the party, everyone must shake hands with everyone whom they do not already know. At the end of the party, I can't remember the number of people I have met, so I go ask all other people at the party how many hands they have shaken, and they each tell me a different number. You do not shake hands with yourself or your 'partner' (the person with whom you came). How many hands did my roommate shake?

Answer: There are 105 handshakes that his roommate shake.Step-by-step explanation:Since we have given that Number of couples = 15As we know the "Handshaking lemma"Here, n = 15So, the number of handshakes would be [tex]\dfrac{n(n-1)}{2}\\\\=\dfrac{15(15-1)}{2}\\\\=\dfrac{15\times 14}{2}\\\\=15\times 7\\\\=105[/tex]Hence, there are 105 handshakes that his roommate shake.