The most popular mathematician in the world is throwing aparty for all of his friends. As a way to kick things off, they decidethat everyone should shake hands. Assuming all 10 people atthe party each shake hands with every other person (but notthemselves, obviously) exactly once, how many handshakes takeplace?

Answer:The no. of possible handshakes takes place are 45.Step-by-step explanation:Given : There are 10 people in the party .To Find: Assuming all 10 people at the party each shake hands with every other person (but not themselves, obviously) exactly once, how many handshakes take place?Solution:We are given that there are 10 people in the partyNo. of people involved in one handshake = 2To find the no. of possible handshakes between 10 people we will use combination over here Formula : [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]n = 10 r= 2Substitute the values in the formula [tex]^{10}C_{2}=\frac{10!}{2!(10-2)!}[/tex][tex]^{10}C_{2}=\frac{10!}{2!(8)!}[/tex][tex]^{10}C_{2}=\frac{10 \times 9 \times 8!}{2!(8)!}[/tex][tex]^{10}C_{2}=\frac{10 \times 9 }{2 \times 1}[/tex][tex]^{10}C_{2}=45[/tex]No. of possible handshakes are 45Hence The no. of possible handshakes takes place are 45.