Chapter 13.2, Problem 18PPS

To determine

To find: thetotal lines that are randomly selected from $10$ and $3$ of which are collinear.

Answer to Problem 18PPS

The total line is $45$ .

Explanation of Solution

Given:

Out of $10$ randomly selected point $3$ which are collinear.

Concept used:

Number of different permutations of $n$ objects where there are $n_1$ repeated items, $n_2$ repeated items,…. $n_k$ repeated items.

  $\frac{n!}{n_1!n_2!\mathrm{...}{n}_k!}$ .

Calculation:

Two point determine a line. If no $3$ points are collinear, then each line will consistof exactly $2$ points. Since, there are $10$ total points, the number of lines is the combination of 10 points taken $2$ at a time, which is

  $\begin{array}{l}\frac{10!}{8!2!}.\\ \Rightarrow \frac{10\times 9\times 8!}{8!2\times 1}=\frac{90}{2}=45.\end{array}$

Hence, the total line is $45$ .