Chapter 8.5, Problem 73E

To determine

To find: the number of diagonals of the polygon.

Answer to Problem 73E

The number of diagonals of the polygon is 5.

Explanation of Solution

Given Information:

A line segment connection any two nonadjacent vertices is called a diagonal of a polygon.

Pentagon

Calculation:

The number of combinations of $n$ elements taken $r$ at a time is

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

To make a diagonal, need to choose a pair out of the 5 vertices of the pentagon. One pair will define one diagonal. If we ciunt number of pairs out of 5 vertices, that will also count the number of sides of the pentagon. So to consider this, subtract the number of sides of the polygon.

The number of diagonal is given by number of pairs out of 5 vertices minus the number of sides.

  $\begin{array}{c}{}_5{C}_2-5=\frac{5!}{\left(5-2\right)!2!}-5\\ =\frac{5!}{3!2!}-5\\ =5\times 2-5\\ =10-5\\ =5\end{array}$