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Using 3 of the points in the set { (x, y) | x, y are integers, 1 ≤ x ≤3, and 3 ≤ y ≤ 10}, how many triangles can be formed?

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There are two ways to tackle this problem:

 

Method 1: Counting Valid Triangles

 

Total Points: We have 3 choices for x (1, 2, or 3) and 8 choices for y (3, 4, 5, ..., 10), resulting in a total of 3 * 8 = 24 points.

 

Choosing 3 Points: We need to choose 3 out of these 24 points to form a triangle.

 

The number of ways to do this can be calculated using combinations: nCr = n! / (r! * (n-r)!) where n is the total number of elements (24) and r is the number of elements to choose (3). 24C3 = 24! / (3! * (24 - 3)!) = 2280

 

Not All Triangles are Valid: However, not all combinations of 3 points will form valid triangles.\

 

The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side. We need to check how many of the 2280 combinations violate this rule.

 

Loop through each combination of 3 points.

 

Calculate the distances between all three pairs of points using the distance formula (square root of the sum of squared differences in x and y coordinates).

 

If any of the distances violate the triangle inequality (i.e., the sum of two distances is less than the third distance), discard that combination.

Valid Triangles: After checking all combinations, count the remaining ones that satisfy the triangle inequality. This will be the number of valid triangles.

 

Method 2: Faster Approach (Counting Invalid Triangles)

 

Total Combinations: Same as method 1, there are 2280 total ways to choose 3 points from 24.

 

Invalid Triangles: We can count the number of triangles that violate the triangle inequality and subtract them from the total to get the valid ones. There are two main cases for invalid triangles:

 

All points on a straight line: If all three chosen points fall on the same line (e.g., (1, 3), (2, 3), (3, 3)), they cannot form a triangle.

 

Degenerate Triangle: If two points have the same x-coordinate or the same y-coordinate, and the third point does not lie above the line connecting them, they cannot form a valid triangle. (Imagine a straight line segment connecting the two points with the same coordinate. The third point must be above this line segment for a triangle to form).

 

Count the number of ways to choose 3 points that fall on the same horizontal or vertical line (3 choices for x-coordinate * 8 choices for y-coordinate = 24)

 

Count the number of ways to choose 2 points with the same x-coordinate (3 choices for x-coordinate * 7 choices for y-coordinate for the first point * 6 choices for y-coordinate for the second point, excluding the chosen y-coordinate = 126).

 

Do the same for points with the same y-coordinate.

 

Valid Triangles: The total number of valid triangles is then: Total Combinations - Invalid Triangles 2280 - (24 + 126 + 126) = 2004

 

Both methods will give you the answer: there are 2004​ triangles that can be formed using 3 of the points.

 Apr 9, 2024

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