Question

At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?

Answer 1

N - the number of people
Everyone shook hands with everybody else, so we must know how many ways there are to choose 2 people from the group of n people. Using combination:
$\binom{n}{2}=\frac{n!}{2!(n-2)!}=\frac{(n-2)! \times (n-1) \times n}{2(n-2)!}=\frac{(n-1)n}{2}$

There were 66 handshakes.
$66=\frac{(n-1)n}{2} \\ 132=(n-1)n \\ 132=n^2-n \\ n^2-n-132=0 \\ n^2+11n-12n-132=0 \\ n(n+11)-12(n+11)=0 \\ (n-12)(n+11)=0 \\ n-12=0 \ \lor \ n+11=0 \\ n=12 \ \lor \ n=-11$

The number of people must be a positive number, so n=12.
There were 12 people at the party.

Answer 2

Every body has two hands so we have to multiply 66 by 2.

66 multiplied by 2 is equal to 132

The answer is 132.

hope it helps.

Can you choose mine as the brainliest answer please please?