Question

six people come to a party and shake hands with each other. if each person shakes the hand of every other person, how many handshakes occur? there will be \\(\square\\) handshakes.

Explanation:

Step1: Understand the problem type

This is a combination problem where we want to find the number of ways to choose 2 people out of 6 to shake hands (since a handshake between person A and person B is the same as between B and A, so order doesn't matter). The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 6 \) (total number of people) and \( k = 2 \) (number of people per handshake).

Step2: Apply the combination formula

First, calculate the factorial values. \( n!=n\times(n - 1)\times\cdots\times1 \), so \( 6! = 6\times5\times4\times3\times2\times1=720 \), \( 2! = 2\times1 = 2 \), and \( (6 - 2)!=4!=4\times3\times2\times1 = 24 \).

Now substitute into the formula: \( C(6, 2)=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!} \).

Simplify the factorials: \( \frac{6\times5\times4!}{2\times1\times4!} \). The \( 4! \) terms cancel out, leaving \( \frac{6\times5}{2\times1} \).

Step3: Calculate the result

\( \frac{6\times5}{2\times1}=\frac{30}{2}=15 \).

Answer:

15