Question:
In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be $70$ then the number of diagonals of the polygon is
In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be $70$ then the number of diagonals of the polygon is
Updated On: Jun 18, 2022
- 20
- 28
- 8
- None of these
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The Correct Option is A
Solution and Explanation
A combination of four vertices is equivalent to one interior points of intersection of diagonals.
$\therefore$ No. of interior points of intersection
$={ }^{n} C_{4}=70$
$\Rightarrow n(n-1)(n-2)(n-3)=5.6 .7 .8$
$\therefore n =8$
So, number of diagonals $={ }^{8} C_{2}-8=20$
$\therefore$ No. of interior points of intersection
$={ }^{n} C_{4}=70$
$\Rightarrow n(n-1)(n-2)(n-3)=5.6 .7 .8$
$\therefore n =8$
So, number of diagonals $={ }^{8} C_{2}-8=20$
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects