Question

How many triangles can be formed by joining the vertices of a hexagon?

Answer 1

Use permutation and combinations to evaluate the required answer:

We need to form triangles by joining the vertices of a hexagon

To form a triangle we require $3$ vertices.

So, from the given $6$ vertices of a hexagon we can choose $3$ vertices in ${}^{6}C_3$ ways

The number of triangles that can be formed$={{}^{6}C}_3$

$=\frac{6!}{3!3!}$ $\left[{}^n{C}_r=\frac{n!}{r!(n-r)!}\right]$

$$\begin{gathered} =\frac{6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1\times 3\times 2\times 1} \\ =20 \end{gathered}$$

Hence, $\mathbf{20}$ triangles can be formed by joining vertices of a hexagon.