Problem Analysis #21

To solve this problem, we need to determine the probability $p_5$ that three randomly chosen distinct points from the given 15 lattice points form a triangle. Here's the step-by-step reasoning:

Step 1: Calculate the total number of ways to choose 3 points from 15.

The number of ways to choose 3 points from 15 is given by the combination formula:

Step 2: Identify collinear triples.

We need to find all sets of 3 points that lie on the same line. The lattice points are in a 3x5 grid. We'll check for collinear triples in horizontal, vertical, and diagonal directions.

Horizontal lines:

• There are 5 horizontal lines (one for each $y$-value from 1 to 5).
• Each horizontal line has 3 points, and the number of ways to choose 3 points from 3 is $\binom{3}{3} = 1$.
• Total horizontal collinear triples: $5 \times 1 = 5$.

Vertical lines:

• There are 3 vertical lines (one for each $x$-value from 1 to 3).
• Each vertical line has 5 points, and the number of ways to choose 3 points from 5 is $\binom{5}{3} = 10$.
• Total vertical collinear triples: $3 \times 10 = 30$.

Diagonal lines:

• Diagonals with slope 1 (from top-left to bottom-right):
  • $(1,1), (2,2), (3,3)$: 1 triple.
  • $(1,2), (2,3), (3,4)$: 1 triple.
  • $(1,3), (2,4), (3,5)$: 1 triple.
  • Total: 3 triples.
• Diagonals with slope -1 (from top-right to bottom-left):
  • $(1,5), (2,4), (3,3)$: 1 triple.
  • $(1,4), (2,3), (3,2)$: 1 triple.
  • Total: 2 triples.

Total collinear triples:

Step 3: Calculate the probability $p_5$.

The probability that the three points are collinear is the ratio of collinear triples to the total number of triples:

Step 4: Simplify the fraction.

The fraction $\frac{40}{455}$ simplifies by dividing both numerator and denominator by their greatest common divisor, which is 5:

Step 5: Find $p + q$.

Here, $p = 8$ and $q = 91$, so:

Final Answer: