If in a regular polygon the number of diagonals is $54$, then the number of sides of this polygon is :

Question

Number of seven digits numbers which can be formed by using all the digits 1, 2, 3, 4, 5 with at least one digit repeated is _____.

Answer 1

To determine the number of sides of a regular polygon given that the number of diagonals is $54$, we use the formula for calculating the number of diagonals in a polygon. The formula is:

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

where $D$ represents the number of diagonals and $n$ represents the number of sides of the polygon.

We are given that $D = 54$. So, we can set up the equation:

\frac{n(n-3)}{2} = 54

First, eliminate the fraction by multiplying both sides by 2:

n(n-3) = 108

Next, solve this quadratic equation. Expand the equation:

n^2 - 3n = 108

Rearrange it to form a standard quadratic equation:

n^2 - 3n - 108 = 0

Now, factorize the quadratic equation:

(n - 12)(n + 9) = 0

From this, we get two possible solutions for $n$:

n - 12 = 0 \quad \Rightarrow \quad n = 12

n + 9 = 0 \quad \Rightarrow \quad n = -9

Since the number of sides cannot be negative, we discard $n = -9$. Therefore, the number of sides of the polygon is $n = 12$.

Thus, the correct answer is: 12

Answer 2