# Difference between revisions of "2019 AMC 10A Problems/Problem 14"

The following problem is from both the 2019 AMC 10A #14 and 2019 AMC 12A #8, so both problems redirect to this page.

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

## Solution 1(David C)

We do casework to find values that work

Case 1: Four Parallel Line= 0 Intersections

Case 2: Three Parallel Lines and One Line Intersecting the Three Lines= 3 Intersections

Case 3: Two Parallel Lines with another Two Parallel Lines= 4 Intersections

Case 4: Two Parallel Lines with Two Other Non-Parallel Lines=5 Intersections

Case 5: Four Non-Parallel Lines All Intersecting Each Other at different points = 6 Intersections

Case 6: Four Non-Parallel Lines All Intersecting At One Point= 1 Intersection

You can find out that you cannot have 2 Intersections

Sum= $1+3+4+5+6$= $19 \boxed{D}$