Question

There are $n$ distinct points on the circumference of a circle. The number of pentagons that can be formed with these points as vertices is equal to the number of possible triangles. Then the value of $n$ is

• A. $7$
• B. $8$
• C. $15$
• D. $30$

Answer 1

The correct option is B

$8$

Explanation of the correct option.

Compute the required value.

Given: There are $n$ distinct points on the circumference of a circle.

We know that for the pentagon we need $5$ points as it's vertices and for the triangle we need $3$ points as it's vertices.

As per given condition, number of pentagons formed with $n$points$=$number of triangles formed with $n$points

Since, combination is the choice of $r$ things from a set of $n$ things without replacement and where order does not matter, combination is given by ${}^{n}C_r=\frac{n!}{r!(n-r)!}$

$$\begin{gathered} {}^{n}C_5={}^{n}C_3 \\ \Rightarrow \frac{n!}{5!(n-5)!}=\frac{n!}{3!(n-3)!} \\ \Rightarrow 3!(n-3)!=5!(n-5)! \\ \Rightarrow 3!(n-3)\times \left(n-4\right)\times (n-5)!=5\times 4\times 3! \\ \Rightarrow (n-3)\left(n-4\right)=20 \\ \Rightarrow n^2-4n-3n+12-20=0 \\ \Rightarrow n^2-7n-8=0 \\ \Rightarrow n^2-8n+n-8=0 \\ \Rightarrow n(n-8)+1(n-8)=0 \\ \Rightarrow (n-8)\left(n+1\right)=0 \end{gathered}$$

So the values of $n$ are $8$ and $-1$.

Since number of points can't be negative thus, $n=8$.

Hence option (B) is the correct option.