In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon is

Options

A. `5//12`
B. `7//12`
C. `2//5`
D. none of these

Explanation

To find the probability that two randomly drawn diagonals of a convex hexagon intersect at an interior point, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of vertices in the hexagon**:
A convex hexagon has 6 vertices.

2. **Calculate the number of diagonals in the hexagon**:
The formula for the number of diagonals in an n-sided polygon is given by:
\[ \text{Number of diagonals} = \binom{n}{2} - n \]
For a hexagon (n = 6):
\[ \text{Number of diagonals} = \binom{6}{2} - 6 \]
Calculating \(\binom{6}{2}\):
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
Therefore, the number of diagonals is:
\[ 15 - 6 = 9 \]

3. **Determine the total ways to choose 2 diagonals**:
The total ways to choose 2 diagonals from the 9 diagonals is given by:
\[ \binom{9}{2} \]
Calculating \(\binom{9}{2}\):
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

4. **Find the number of ways to choose 4 vertices**:
For two diagonals to intersect at an interior point, we need to choose 4 vertices from the 6 vertices of the hexagon. The number of ways to choose 4 vertices is:
\[ \binom{6}{4} \]
Since \(\binom{6}{4} = \binom{6}{2}\), we already calculated:
\[ \binom{6}{4} = 15 \]

5. **Calculate the probability**:
The probability that two diagonals intersect at an interior point is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{intersect}) = \frac{\text{Number of ways to choose 4 vertices}}{\text{Total ways to choose 2 diagonals}} = \frac{15}{36} \]
Simplifying \(\frac{15}{36}\):
\[ P(\text{intersect}) = \frac{5}{12} \]

### Final Answer:
The probability that the diagonals intersect at an interior point of the hexagon is \(\frac{5}{12}\).

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