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Difficulty: HardCombinations

A plane contains 1212 distinct points such that exactly 55 of them lie on the same straight line, while no other subset of three points is collinear. How many distinct straight lines can be formed by joining pairs of these points?

Answer: 57 lines

Answer

The total number of distinct straight lines that can be formed is 57.
The total number of distinct straight lines is obtained by taking all possible pairs of points from 12, which is 12C2=66^{12}C_2 = 66, subtracting the pairs formed among the 5 collinear points (5C2=10^5C_2 = 10), and adding 1 back to account for the single straight line on which those 5 points lie. This gives 66−10+1=5766 - 10 + 1 = 57.

Step-by-Step Solution

1
Calculate the total possible lines by selecting any 2 points out of 12.
12C2=12×112=66^{12}C_2 = \frac{12 \times 11}{2} = 66
Any pair of non-collinear points uniquely determines a straight line.
2
Calculate the combinations of 2 points selected from the 5 collinear points.
5C2=5×42=10^5C_2 = \frac{5 \times 4}{2} = 10
These 10 point pairs all lie on the exact same single straight line instead of forming 10 distinct lines.
3
Adjust the count to find the net number of distinct straight lines.
66−10+1=5766 - 10 + 1 = 57
Subtracting the 10 overcounted combinations and adding back 1 for the single line passing through all 5 collinear points gives the correct total.

Key Concept

Combinations with Collinear Constraints
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Topics

Combinations

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