Consider a polygon of sides 'n' which satisfies the equation `3*.^(n)P_(4)=.^(n-1)P_(5)`.
Q. Number of quadrilaterals that can be made using the vertices of the polygon of sides 'n' if exactly two adjacent sides of the quadrilateral are common to the sides of the n-gon is

Options

A. 50
B. 60
C. 70
D. 80

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Understand the Given Equation
The equation provided is:
\[ 3 \cdot nP_4 = n-1P_5 \]

### Step 2: Expand the Permutations
Using the formula for permutations, we can express \( nP_r \) as:
\[ nP_r = \frac{n!}{(n-r)!} \]
Thus, we can rewrite the equation as:
\[ 3 \cdot \frac{n!}{(n-4)!} = \frac{(n-1)!}{(n-6)!} \]

### Step 3: Simplify the Equation
We can simplify this further:
\[ 3 \cdot n \cdot (n-1)(n-2)(n-3) = (n-1)(n-2)(n-3)(n-4)(n-5) \]
Now, cancel out the common terms:
\[ 3n = (n-4)(n-5) \]

### Step 4: Expand and Rearrange
Expanding the right side:
\[ 3n = n^2 - 9n + 20 \]
Rearranging gives us:
\[ n^2 - 12n + 20 = 0 \]

### Step 5: Solve the Quadratic Equation
We can factor or use the quadratic formula to solve for \( n \):
\[ n^2 - 12n + 20 = 0 \]
Using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -12, c = 20 \):
\[ n = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} \]
\[ n = \frac{12 \pm \sqrt{144 - 80}}{2} \]
\[ n = \frac{12 \pm \sqrt{64}}{2} \]
\[ n = \frac{12 \pm 8}{2} \]
This gives us two possible values:
\[ n = 10 \quad \text{or} \quad n = 2 \]

Since \( n \) represents the number of sides in a polygon, it cannot be 2. Therefore, we have:
\[ n = 10 \]

### Step 6: Calculate the Number of Quadrilaterals
Now, we need to find the number of quadrilaterals that can be formed using the vertices of the polygon with 10 sides, where exactly two adjacent sides of the quadrilateral are common to the sides of the polygon.

### Step 7: Choose Two Adjacent Sides
We can choose any two adjacent sides from the 10 sides of the polygon. The number of ways to choose two adjacent sides is equal to the number of sides, which is 10.

### Step 8: Select the Fourth Vertex
After selecting two adjacent sides (let's say sides 1 and 2), we need to select a fourth vertex from the remaining vertices. Since two vertices are already used, we have 8 vertices left to choose from.

### Step 9: Total Combinations
Thus, the total number of quadrilaterals is:
\[ \text{Total Quadrilaterals} = 10 \times 8 = 80 \]

### Final Answer
The number of quadrilaterals that can be formed is **80**.

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