Question

There are 12 friends. Each greeted the other by shaking hands. How many handshakes took place?

Answer 1

In this question we have been given with the data that there are $12$ friends who greet each other by shaking of hands. We will solve this question by considering the number of people to be $n$. We will then use the combination formula given as $^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$ which includes all the $12$ friends shaking hands with each other without repetition and given that a person can’t shake hands with himself.

Complete step-by-step solution:
We know that there are a total of $12$ friends and each of them shake hands with each other. Let the number of friends who shake hands be $n$.
We know that a person will shake hands with everyone but himself therefore we do not want repetition in the number of handshakes.
We also know that at a time $2$ people shake hands with each other therefore, we have the total number of people as $n=12$ and $r=2$.
On substituting the values of $n$ and $r$ in the combination formula, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=\dfrac{12!}{\left( 12-2 \right)!2!}$
now on simplifying the terms in the denominator, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=\dfrac{12!}{10!2!}$
Now, we can write $12!=12\times 11\times 10!$ therefore, on substituting, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=\dfrac{12\times 11\times 10!}{10!2!}$
On simplifying, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=\dfrac{12\times 11}{2!}$
On writing $2!=2\times 1$, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=\dfrac{12\times 11}{2\times 1}$
On simplifying the terms, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=6\times 11$
On multiplying the terms, we get:
${{\Rightarrow }^{12}}{{C}_{2}}=66$, which is the total number of handshakes that will occur when $12$ friends greet each other, which is the required solution.

Note: It is to be remembered that in these types of questions we use the combination formula because unlike the permutation formula, the order of the selection does not matter. The general formula should be remembered for finding the number of handshakes between $n$ people which is given as $\dfrac{n\left( n-1 \right)}{2}$.