There are lots of ways of choosing three dots from this 4 by 4 array. How many triples of points are there when all three lie on a straight line (not necessarily equally spaced)
40
To find the number of triples of points that lie on a straight line in a 4 by 4 array, we can consider the different possible types of lines they can form - horizontal lines, vertical lines, and diagonal lines.
1. Horizontal lines:
In a 4 by 4 array, there are 4 rows. For each row, we can select any 3 points from that row to form a horizontal line. So, for each row, there are (4 choose 3) = 4 triplets of points. Since there are 4 rows, the total number of triplets for horizontal lines is 4 * 4 = 16.
2. Vertical lines:
In a 4 by 4 array, there are 4 columns. For each column, we can select any 3 points from that column to form a vertical line. Similar to the previous case, there are (4 choose 3) = 4 triplets of points for each column. Thus, the total number of triplets for vertical lines is again 4 * 4 = 16.
3. Diagonal lines:
There are two types of diagonal lines in a 4 by 4 array - main diagonal (from top left to bottom right) and off-diagonal (from top right to bottom left).
For the main diagonal, we can select any 3 points from the main diagonal to form a line. Since the main diagonal contains 4 points, the number of triplets for the main diagonal is (4 choose 3) = 4.
For the off-diagonal, we can select any 3 points from the off-diagonal to form a line. Similarly, the number of triplets for the off-diagonal is also (4 choose 3) = 4.
Finally, we sum up the number of triplets for each type of line:
Horizontal lines: 16
Vertical lines: 16
Main diagonal: 4
Off-diagonal: 4
Total number of triplets on a straight line: 16 + 16 + 4 + 4 = 40.
So, there are 40 triples of points in the 4 by 4 array when all three points lie on a straight line.