practice problems

Q1. Straight lines are drawn by joining (m) points on a straight line to (n) points on another line. No two lines drawn are parallel and no three lines are concurrent. How many total intersecting points are there?
S1.

Let's rephrase the problem as how many new intersection points are created?

Solution 1:
Let's create a quadrilateral by joining 2 vertices from each line.
In this quadrilateral, 2 new intersection points are created.
One by diagonals.
And one by extending the 2 sides which we just created.

Total ways to create a quadrilateral like this:
Choose 2 vertices from one line and 2 from another.
So,
answer = 2 * mC2 * nC2

Solution 2:
Total lines: mn
Total intersection points: (mn)C(2)
From this we need to subtract the intersection points which were already there.
On each of the 'm' points, 'n' lines converge.
So nC2 intersection points are made by them on a single point.
In total m*nC2 such points.
Similarly n*mC2 other points.

So mnC2 - m*nC2 - n*mC2
= [mn.(mn-1) - mn(n-1) - nm(m-1)]/2
= mn/2[mn - 1 - n + 1 -m + 1]
= mn/2[mn -m -n +1] = mn/2[(m-1)(n-1] = 2 * mC2 * nC2




Q2. Show that the number of rectangles on any chess board of size (n x n) is [n * (n+1)/2]^2
S2.
So it is (n+1)C2 * (n+1)C2.
Since you have to choose 2 horizontal lines and 2 vertical lines from (n+1) horizontal and (n+1) vertical lines.