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

Here in this question we have been asked to find the value of $n$ when the information is given as “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.” We know that the number of pentagons can be made using $n$ distinct points is given as $^{n}{{C}_{5}}$ and similarly for triangles it is given as $^{n}{{C}_{3}}$ .

Complete step-by-step solution:
Now considering from the question we have been asked to find the value of $n$ when the information is given as “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.”
From the basic concepts we know that the number of pentagons can be made using $n$ distinct points is given as $^{n}{{C}_{5}}$ and similarly for triangles it is given as $^{n}{{C}_{3}}$ .
From the given information we can equate both after that we will have $^{n}{{C}_{5}}{{=}^{n}}{{C}_{3}}$ .
From the basic concepts we know that $^{n}{{C}_{r}}=\dfrac{\left( n \right)!}{r!\left( n-r \right)!}$ .
Now we can say that $\Rightarrow \dfrac{n!}{3!\left( n-3 \right)!}=\dfrac{n!}{5!\left( n-5 \right)!}$ .
Now we will further simplify this and write it as
$\begin{aligned} & \Rightarrow 3!\left( n-3 \right)!=5!\left( n-5 \right)! \\\ & \Rightarrow \left( n-3 \right)!=\left( 5\times 4 \right)\left( n-5 \right)! \\\ & \Rightarrow \left( n-3 \right)\left( n-4 \right)=20 \\\ \end{aligned}$
Now if we try substituting the options in this equation and verify then it will satisfy only for option “B”. Hence we can conclude that the value of $n$ is $8$ . Hence we can mark the option “B” as correct.

Note: During this process of answering this type of questions we should be sure with our concepts that we apply and the calculations that we perform. Alternatively without substituting the options we can simplify the equation be writing or expressing it as $\begin{aligned} & \Rightarrow \left( n-3 \right)\left( n-4 \right)=5\times 4 \\\ & \Rightarrow n-3=5 \\\ & \Rightarrow n=8 \\\ \end{aligned}$ .