Question

In a polygon, no three diagonals are concurrent . If the total number of points of intersection of diagonals interior to the polygon is \[70\], the number of the diagonals of the polygon is
A.\[20\]
B.\[28\]
C.\[8\]
D.None of these

Answer 1

Hint: In the given question, we need to find the number of diagonals of the polygon. A diagonal is nothing but a line joining the two vertices of the polygon.The bounded plane region is known as a polygon. Simple polygons don't intersect each other. The point where the two edges meet is known as polygon vertices. So in order to solve this problem we’ll take a selection of four vertices of the polygon because of the given interior intersection. Now we’ ll find the number of sides “n” with the help of \[ {^n{C}}{_4} = 70\] as we have given that the total number of points of intersection of diagonals interior to the polygon is \[70\].

Complete answer:

A selection of four vertices of the polygon given an interior intersection.
The number of sides = \[n\]
\[ {^n{C}}{_4} = 70\]
By expanding,
We get,
\[\dfrac{n\left( n – 1 \right)\left( n – 2 \right)\left( n – 3 \right)}{4.3.2.1} = \ 70\]
We need to find the value of \[n\]
\[\dfrac{n\left( n – 1 \right)\left( n – 2 \right)\left( n – 3 \right)}{24} = 70\]
\[n\left( n – 1 \right)\left( n – 2 \right)\left( n – 3 \right) = 70 \times 24\]
By multiplying,
We get,
 \[n^{4} – 6n^{3} + 11n^{2} – 6n = 1680\]
Using synthetic division, we can write polynomial completely as follows,
\[\left( n – 8 \right)\left( n + 5 \right)\left( n^{2} – 3n + 42 \right) = 0\]
Here only\[\ n = \ 8\ \] is applicable to this equation.
 \[n = 8\]
Thus the polygon has \[8\] sides.
The number of diagonals =\[ {^8{C}}{_2} - 8\]
 By expanding,
We get,
\[= \dfrac{8!}{2!\left( 8 - 2 \right)!} - 8\]
\[= \dfrac{8!}{2!6!} - 8\]
On further expanding,
\[= \dfrac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} - 8\]
By simplifying,
We get,
\[= 28 - 8 = 20\]
The number of diagonals = \[ \ 20\]
Final answer :
The number of diagonals = \[ \ 20\]
Hence the correct answer is A.

Note:
We should not confuse combinations and permutations. Combination is the technique that determines the number of possible arrangements in the collection of items. Permutations is the selection process and defined as the number of ways of arranging items in a particular order. In other words, selection of subsets is known as permutation whereas the non fraction order of selection is known as combination.