Q113 / 180 · COMEDK 2023 (Memory Based) · Mathematics

A polygon of $n$ sides has 105 diagonals, then $n$ is equal to

  • A
    20
  • B
    21
  • Correct answer
    C
    15
  • D
    -14

Solution

$\because$ The total number of lines joining any two points of the polygon is given by ${ }^n C_2$ So, ${ }^n C_2=105$ $\begin{aligned} & \Rightarrow \quad \frac{n(n-1)}{2}=105 \\ & \Rightarrow \quad n^2-n=210 \\ & \Rightarrow \quad n^2-n-210=0 \\ & \Rightarrow \quad n^2-15 n+14 n-210=0 \\ & \Rightarrow \quad n(n-15)+14(n-15)=0 \\ & \end{aligned}$ $\begin{aligned} & \Rightarrow \quad(n-15)(n+14)=0 \\ & \text { either } n-15=0 \text { or } n+14=0 \\ & \Rightarrow \quad n=15 \text { or }-14 \\ & \because \text { Number of sides cannot be negative } \\ & \therefore n=15\end{aligned}$

Hence, the correct answer is option: 'C'.

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