Chapter 13.5, Problem 45E

To determine

To calculate: The number of different ways to form 4 sided figure by joining 4 points.

Answer to Problem 45E

The number of different ways to form 4 sided figure by joining 4 pointsare $126$ .

Explanation of Solution

Given information:

There are 9 points on a circle and 4 sides closed figure has to be formed by joining any of 4 points.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there are 9 points on a circle and 4 sides closed figure has to be formed by joining any of 4 points.

In order to form 4 sided closed figure, use combination as selection does not matter.

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Required number of ways are $C\left(9,4\right)$

Here n is 9 and r is 4.

  $\begin{array}{c}C\left(9,4\right)=\frac{9!}{4!\left(9-4\right)!}\\ =\frac{9\cdot 8\cdot 7\cdot 6\cdot 5!}{4!\cdot 5!}\\ =\frac{9\cdot 8\cdot 7\cdot 6}{4\cdot 3\cdot 2\cdot 1}\\ =126\end{array}$

Thus, the number of different ways to form 4 sided figure by joining 4 points are $126$ .