In a class of origami, in Japan, my teacher Yamamotoyama gave me a circular paper, a knife and a scissors. She asked me to cut the paper in order to make triangles out of the circular sheet. She laid down some rules which were mandatory to follow for every origami student.
A sheet of paper has n points marked on its circumference
One must cut through the marked points only
Any cut must last from one marked point to another marked point
One has to cut the paper as many times as the number of chords are possible with n points on the circumference of a circle
One cannot displace the piece until all the possible cuts are made
Find the maximum possible number of triangles which do not have any of its vertices out of the n marked points

Options

A. 0
B. 3
C. `^nC_3`
D. `^nC_6`

Answer

To solve the problem of finding the maximum possible number of triangles that can be formed from n marked points on the circumference of a circle, we can follow these steps:

### Step 1: Understand the Problem
We need to create triangles using the marked points on the circumference of the circle. Each triangle must have its vertices at these marked points, and we can only cut through the marked points.

**Hint:** Visualize the circle with n points and think about how triangles can be formed using these points.

### Step 2: Identify the Requirements for Forming a Triangle
To form a triangle, we need to connect three points. However, the problem specifies that we need to ensure that the triangles do not have any of their vertices outside the marked points.

**Hint:** Remember that a triangle is defined by three points, and all points must be on the circumference.

### Step 3: Determine the Minimum Points Needed
To form a triangle, we need at least 3 points. However, to create a triangle that does not extend outside the marked points, we need to consider the chords formed by these points.

**Hint:** Think about how many points are needed to create enough chords to form a triangle.

### Step 4: Analyze the Chords
To create a triangle inside the circle, we need to have at least 3 chords. Each chord connects two points. To form 3 chords, we need at least 6 points. This is because each chord requires two points, and three chords would require a minimum of 6 points.

**Hint:** Count the number of points needed for the chords that will form the triangle.

### Step 5: Calculate the Number of Ways to Choose Points
Now that we know we need at least 6 points to form a triangle, we can calculate the number of ways to choose 6 points from n marked points. The number of ways to choose k points from n points is given by the combination formula \( nCk \).

**Hint:** Use the combination formula \( nCk = \frac{n!}{k!(n-k)!} \) to find the number of combinations.

### Step 6: Final Calculation
Since we need to choose 6 points from n, the number of triangles that can be formed is given by \( nC6 \).

**Hint:** Remember that the answer is based on the number of ways to select the points, not the specific points themselves.

### Conclusion
The maximum possible number of triangles that can be formed with n marked points on the circumference of a circle is given by \( nC6 \).

**Final Answer:** The answer is option D: \( nC6 \).