In a plane there are 10 points, of which 5 are collinear and the other 5 are in general position (no extra collinearities). How many distinct straight lines can be formed by joining pairs of these points?

Introduction / Context:
Lines determined by a finite point set are counted by considering all pairs, but collinear points collapse many pairs into the same line. We correct the overcount caused by the 5 collinear points.

Given Data / Assumptions:

• Total points = 10.
• A single set of 5 collinear points; remaining 5 are in general position and not on that line.

Concept / Approach:
Start with all pairwise lines C(10,2), then adjust for the collinear set. Among the 5 collinear points, C(5,2)=10 pairs lie on the same single line; these 10 pairs would naively count as 10 different lines but yield only 1. Therefore, subtract 9 from the naive total.

Step-by-Step Solution:
Naive count = C(10,2) = 45.Within the 5 collinear points: 10 pairs but only 1 line ⇒ overcount = 10 − 1 = 9.Corrected total = 45 − 9 = 36.

Verification / Alternative check:
Count lines through the 5 collinear points (1), lines from each of the other 5 points to the collinear set (all distinct), and lines among the other 5—this decomposition matches 36.

Why Other Options Are Wrong:
45 ignores the collinearity; 35 and 30 over-subtract.

Common Pitfalls:
Subtracting 10 instead of 9 (remember one line remains).

Final Answer:
36