6.3 Q-10 | Permutation, Combination And Probability | Mathematics 11

Exercise 6.3 — Question 10

Find the number of diagonals of a polygon having $n$ sides. Also find the number of diagonals of a hexagon.

A diagonal of a polygon is a line segment joining two non-adjacent vertices.

Total number of line segments joining any 2 vertices out of $n$ vertices: $^{n} C_{2} = \frac{n ( n - 1 )}{2}$
This count includes the $n$ sides of the polygon.
Therefore, the number of diagonals is: $Diagonals =^{n} C_{2} - n = \frac{n ( n - 1 )}{2} - n = \frac{n ( n - 1 ) - 2 n}{2} = \frac{n ^{2} - 3 n}{2} = \frac{n ( n - 3 )}{2}$

General Formula: $Number of diagonals = \frac{n ( n - 3 )}{2}$

For a Hexagon ( $n = 6$ ): $Diagonals = \frac{6 ( 6 - 3 )}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9$

A hexagon has 9 diagonals.

When solving equations of the form $^{n} C_{p} =^{n} C_{q}$ , use the property:

$^{n} C_{p} =^{n} C_{q} ⟹ p = q or p + q = n$

Example: If $^{n} C_{15} =^{n} C_{7}$ , then since $15 \neq = 7$ : $n = 15 + 7 = 22$