Let p_n denote the total number of triangles | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The number of triangles $p_n$ formed by joining the vertices of an $n$-sided polygon is given by the combination formula $\binom{n}{3}$.Given the

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(C) The number of triangles $p_n$ formed by joining the vertices of an $n$-sided polygon is given by the combination formula $\binom{n}{3}$.Given the relation $p_{n+1} - p_n = 66$, we substitute the formula: $\binom{n+1}{3} - \binom{n}{3} = 66$.Using the property $\binom{n+1}{k} - \binom{n}{k} = \binom{n}{k-1}$, we get $\binom{n}{2} = 66$.Expanding the combination: $\frac{n(n-1)}{2} = 66$, which simplifies to $n^2 - n = 132$.Rearranging gives the quadratic equation $n^2 - n - 132 = 0$.Factoring the equation: $(n - 12)(n + 11) = 0$.Since $n$ must be positive, $n = 12$.The prime factorization of $12$ is $2^2 	imes 3^1$.The distinct prime divisors of $12$ are $2$ and $3$.The sum of these prime divisors is $2 + 3 = 5$.

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-sided regular polygon. If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is:

Similar Questions

$A$ polygon has $44$ diagonals. Then the number of sides of the polygon is:

Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

$A$ regular polygon has $170$ diagonals. Then the measure of the interior angle of the polygon is

In a row,$6$ boys and $6$ girls are arranged at random. Find the probability that all $6$ girls are together.

There were two women participating in a chess tournament. Every participant played two games with every other participant. The number of games that the men played between themselves exceeded the number of games that the men played with the women by $66$. The total number of participants is

Vedclass Test Series

Exam Paper Generator

Online Exam Module