Question

There are 16 points in a plane, no three of which are in a straight line except 8 which are all in a straight line. The number of triangles that can be formed by joining them equals
A. $504$
B. $552$
C. $560$
D. $1120$

Answer 1

Hint: We need at least three points to form a triangle. We cannot form a triangle if points lie on a straight line i.e., if points are collinear. If there are n points out of which m are collinear then the number of triangles formed is given by the formula ${}^n{C_3} - {}^m{C_3}$. By substituting the total number of points in the plane and the number of collinear points out of the total points, we can obtain the number of triangles.

Complete step by step solution:
In order to form a triangle, we need at least three points.
The number of the triangles formed by the 16 points is ${}^{16}{C_3}$
We cannot form a triangle if points lie on a straight line.
Here 8 points lie on the straight line.
Therefore, we cannot form a triangle with the given 8 points.
Hence the number of triangles that can be formed = ${}^{16}{C_3} - {}^{8}{C_3}$
                                                                                          = $\dfrac{{16!}}{{3!(16 - 3)!}} - \dfrac{{8!}}{{3!(8 - 3)!}}$
                                                                                          = $\dfrac{{16!}}{{3!13!}} - \dfrac{{8!}}{{3!5!}}$
                                                                                          = $\dfrac{{16 \times 15 \times 14}}{{3 \times 2}} - \dfrac{{8 \times 7 \times 6}}{{3 \times 2}}$
                                                                                           = $\dfrac{{3360 - 336}}{6}$
                                                                                           = $\dfrac{{3024}}{6}$
                                                                                           = $504$

Option ‘A’ is correct

Note: In order to solve the given question, one must know to form and calculate combinations. We need at least three points to form a triangle and a triangle cannot be formed if the points given are collinear. The number of ways of selecting r objects from n objects is given by ${}^{n}{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ where $n! = n \times (n - 1) \times (n - 2) \times ...... \times 3 \times 2 \times 1$.