Consider n points in a plane no three of which are collinear and the ratio of number of hexagon and octagon that can be formed from these n points is 4:13 then find the value of n.

Options

A. A. 33
B. B. 41
C. C. 20
D. D. none of these

Answer

To solve the problem of finding the value of \( n \) given that the ratio of the number of hexagons to octagons that can be formed from \( n \) points in a plane (where no three points are collinear) is \( 4:13 \), we can follow these steps:

### Step-by-step Solution:

1. **Understand the Combinations**:
- The number of ways to choose \( r \) points from \( n \) points is given by the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
- For a hexagon, we need to choose 6 points, so the number of hexagons is \( \binom{n}{6} \).
- For an octagon, we need to choose 8 points, so the number of octagons is \( \binom{n}{8} \).

2. **Set Up the Ratio**:
- According to the problem, the ratio of the number of hexagons to octagons is \( \frac{\binom{n}{6}}{\binom{n}{8}} = \frac{4}{13} \).

3. **Express the Combinations**:
- We can express the combinations:
\[ \frac{\binom{n}{6}}{\binom{n}{8}} = \frac{\frac{n!}{6!(n-6)!}}{\frac{n!}{8!(n-8)!}} = \frac{8!(n-8)!}{6!(n-6)!} \]
- Simplifying this gives:
\[ \frac{8 \times 7}{(n-6)(n-7)} \]

4. **Set Up the Equation**:
- Now we set up the equation based on the ratio:
\[ \frac{8 \times 7}{(n-6)(n-7)} = \frac{4}{13} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ 13 \times 56 = 4(n-6)(n-7) \]
- Simplifying \( 13 \times 56 \) gives \( 728 \), so:
\[ 728 = 4(n^2 - 13n + 42) \]

6. **Expand and Rearrange**:
- Expanding the right side:
\[ 728 = 4n^2 - 52n + 168 \]
- Rearranging gives:
\[ 4n^2 - 52n + 168 - 728 = 0 \]
- This simplifies to:
\[ 4n^2 - 52n - 560 = 0 \]

7. **Divide by 4**:
- Dividing the entire equation by 4:
\[ n^2 - 13n - 140 = 0 \]

8. **Factor the Quadratic**:
- We can factor this as:
\[ (n - 20)(n + 7) = 0 \]

9. **Solve for \( n \)**:
- This gives us two solutions:
\[ n - 20 = 0 \quad \Rightarrow \quad n = 20 \]
\[ n + 7 = 0 \quad \Rightarrow \quad n = -7 \quad (\text{not valid since } n \text{ must be positive}) \]

10. **Final Answer**:
- Therefore, the value of \( n \) is \( 20 \).