Question

There are 15 lines terminating at a point. The number of angles formed is:
A. 105
B. 120
C. 135
D. 125

Answer 1

We have to find the number of angles formed from 15 lines and a fixed point. An angle is made using 2 lines and a point. Therefore, to find the total number of angles formed, we will use the concept of combination when there are 15 lines and we have to select 2 from them.

Complete step-by-step answer:
If there are 15 lines, this implies there are 15 points.
Also, there is a common point.
In total there are 16 points.
We want to find the number of angles formed by the 15 lines.
It is known that the angle is made between lines having a common point.
Since, the common point is fixed.
Then, an angle between 2 lines needs 2 more points.
That is we have to select 2 points from 15 points.
We will use combinations to find the total number of angles formed.
Thus, the number of angles are,
$^{15}{C_2}$
If we have $^n{C_r}$, then it is equal to $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Thus, $^{15}{C_2} = \dfrac{{15!}}{{2!\left( {15 - 2} \right)!}}$
Also, $n! = n.\left( {n - 1} \right).....3.2.1$
$\dfrac{{15!}}{{2!13!}} = \dfrac{{15.14.13!}}{{2.1.13!}} \\\ = 15\left( 7 \right) \\\ = 105 \\\$
Thus, there are 105 angles formed.
Hence, option A is correct.

Note: When the order of the choices do not matter, then we use combinations to find the required number of ways. But, if the order matters in selection, like when a word is formed, the order of letters play an important role, then we use permutation.