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