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Incentre and Its Construction

There are n c...

Question

There are n concurrent lines and another line parallel to one of them. The number of different triangles that will be formed by the (n+1) lines, is

\frac{(n - 1) n}{2}

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\frac{(n - 1) (n - 2)}{2}

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\frac{n (n + 1)}{2}

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\frac{(n + 1) (n + 2)}{2}

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Solution

The correct option is B $\frac{(n - 1) (n - 2)}{2}$
The number of triangles $=$ Number of selections of $2$ lines from the $(n - 1)$ lines which are cut by the last line

$=^{n - 1} C_{2} = \frac{(n - 1)!}{2! (n - 3)!}$

$= \frac{(n - 1) (n - 2) (n - 3)!}{2! (n - 3)!}$

$= \frac{(n - 1) (n - 2)}{2}$

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