How many straight lines can be formed using 10 points

gmat_guy666 wrote:How many straight lines can be formed using 10 points , 4 of which are collinear ?

A) 20
B) 30
C) 40
D) 50
E) 60

That's a very vague (ambiguous) question.
Do the lines need to pass through any of the points? Is passing through 1 point enough? 2 points? 3 points?

Given that the correct answer is C, I assume the question should be worded as follows:

If a line is defined two points, how many different lines can be formed with 10 points, 4 of which are collinear ?

NOTE: IF none of the points were collinear (all in a row), then the number of lines = 10C2 (10 points, choose 2 of them).

10C2 = 45, so if we IGNORE the fact that some points are collinear, the answer is 45.

HOWEVER, we need to subtract the lines that can be created using 2 of the 4 collinear points, since these lines will all be the SAME.
In how many ways can we select 2 points from the 4 collinear points?
We can do this in 4C2 ways (= 6 ways)
In other words, all 6 possible lines are the SAME line.
So, we have counted 5 extra lines.

45 - 5 = 40

Answer: C

Aside: If anyone is interested, we have a free video on calculating combinations (like 10C2 and 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Cheers,
Brent

Hi gmat_guy666,

Another way to think about this set-up is to 'map out' the options (without physically drawing them all).

I'm going to call the first 6 points A, B, C, D, E and F and the 4 collinear points G,H,I,J

Point A can form a line with any of the other 9 points = 9 lines
Point B can also form a line with any point (but already formed a line with point A, so we can't count that line twice) = 8 lines
Point C already formed lines with Point A and Point B.....= 7 lines
Point D = 6 lines
Point E = 5 lines
Point F = 4 lines

+ 1 more line formed by GHIJ....

9+8+7+6+5+4+1 = 40 lines

Final Answer: C

GMAT assassins aren't born, they're made,
Rich