Chapter 9.1, Problem 52E

Solution

a.

To find: Why the number of segments connecting all pairs of vertices is $C_{2_n}$ .

Given information:

The quadratic regression of a regular polynomial is $\left(n^2-3n\right)/2$ .

Explanation:

The given quadratic regression of a regular polynomial is,

  $\left(n^2-3n\right)/2$

The vertices of the diagonal of the regular polynomial is $n$ .

A regular polygon with $n$ points has $n$ ways to choose two different points from the set of n points that make up the polygon, where $n$ is the total number of segments connecting all pairs of vertices. This number is ${}_n{C}_2$ .

Therefore, the number of segments connecting all pairs of vertices is $C_{2_n}$ .

b.

To prove: The number of diagonals is $\left(n^2-3n\right)/2$ .

Given information:

The quadratic regression of a regular polynomial is $\left(n^2-3n\right)/2$ .

Proof:

The given quadratic regression of a regular polynomial is,

  $\left(n^2-3n\right)/2$

The vertices of the diagonal of the regular polynomial is $n$ .

From part (a) let’s take ${}_n{C}_2$

The expression for ${}_n{C}_2$ is given by,

  $\begin{array}{c}{}_n{C}_2=\frac{n\left(n-1\right)}{2}\\ =\frac{n^2-n}{2}\end{array}$

This is the number of segments.

Subtracting the above expression by $n$ to get the number of diagonals, this is the number of segments on the perimeter.

Thus the number of diagonals is given by,

  $\begin{array}{c}\frac{n^2-n}{2}-n=\frac{n^2-n-2n}{2}\\ =\frac{n^2-3n}{2}\end{array}$

Therefore, the number of diagonals is $\left(n^2-3n\right)/2$ has been proved.