Chapter 13.9, Problem 32PPS

To determine

To Find: Number of people in the party.

Answer to Problem 32PPS

Total number of people in the party if there were $300$ hand shakes is $25$ .

Explanation of Solution

Given information:

There are total $300$ handshakes as each people shook hands with every other person.

Calculation:

Let’s consider, $A$ and $B$ shakes hands with each other.

Since $A$ shaking hands with $B$ is same as $B$ shaking hands with $A$ . Therefore, this is a combination.

Assume the total number of people in the party be $n$ .

There are $300$ handshakes in total.

  $C\left(n,2\right)=300$

After simplification,

  $C\left(n,2\right)=\frac{P\left(n,2\right)}{Totalnumberofcases}$

For calculating $P\left(n,2\right)$ ,

  $\begin{array}{l}P\left(n,2\right)=n\left(n-1\right)\\ C\left(n,2\right)=\frac{n\left(n-1\right)}{2}\\ 300=\frac{n\left(n-1\right)}{2}\\ 600=n\left(n-1\right)\end{array}$

Now simplify for $n$ ,

  $\begin{array}{l}600=n\left(n-1\right)\\ n^2-n-600=0\\ n^2-25n+24n-600=0\\ n\left(n-25\right)+24\left(n-25\right)=0\end{array}$

Further simplification is,

  $\begin{array}{l}n\left(n-25\right)+24\left(n-25\right)=0\\ n-25=0\\ n=25\end{array}$

Hence total number is $25$ .