Mathematics - Permutation & Combination Question with Solution | TestHub

The number of interior intersection points of diagonals in an $n$-sided polygon, with no three diagonals concurrent, is given by the combination formula for choosing 4 vertices:

$$C(n,4)=\left(\genfrac{}{}{0ex}{}{n}{4}\right)$$

We are given that the number of intersections is 70:

$$\frac{n(n-1)(n-2)(n-3)}{24}=70$$

Solving for $n$, we find:

$$n(n-1)(n-2)(n-3)=1680$$

By testing integer values, we find that

$n=8$

is the solution, as

$8\times 7\times 6\times 5=1680$.

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Step 2: Calculate the number of diagonals

The total number of diagonals in an $n$-sided polygon is given by the formula:

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

Using $n=8$ from Step 1:

$$D=\frac{8(8-3)}{2}=\frac{8\times 5}{2}=\frac{40}{2}=20$$

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**Answer:** The number of diagonals of the polygon is **20**.