Counting lines!

This is what is known as a complete graph . This means that every point is connected to every other point - but not itself!

Therefore, if we have 20 points , each with 19 lines extending out from it, we must have $20 \times 19$ lines (that is to say, 20 sets of 19 lines, one set for each point). However, this actually counts each line twice , as each one is shared by two points. We must divide by 2 to get the result.

Therefore our result will be: $20\times 19 \over 2$ $= 190$

To use an analogy, if you want to count the number of handshakes made in a room (with $n$ people) where each person shakes hands once with every other person, if you count all the handshakes made by every person ( $n \times (n - 1)$ ), this will be twice as many as you need, because each handshake is made by 2 people. This is why we would half the total number of handshakes made by everyone, i.e. $n \times (n-1) \over 2$

Label the points as $A , B , C , D \dots\ T$

Point $A$ can be paired with $19$ points

Point $B$ can be paired with $18$ lines. (Since $B$ is already paired with $A$ )

Point $C$ can be paired with $17$ lines.(Since $C$ is already paired with $A , B$ )

$\dots$

$\dots$

Point $S$ can be paired with $1$ lines (Since $S$ is already paired with $A , B , C \dots\ Q , R$ )

Point $T$ can be paired with $0$ lines (Since $T$ is already paired with $ALL!$ )

Hence , total lines $= \displaystyle \sum_{i=1}^{19}$

$=\displaystyle \frac{19 \times (19 + 1)}{2}$

$= \displaystyle\frac{19 \times 20}{2}$

$=\displaystyle 19 \times 10$

$=\Large \boxed{190}$

For generalization, there are n points. To make line, we must connect one point to another point (2 points is required to make a line). So the lines that we can make is ${n \choose 2 }$

Starting at the first point, it must be joined to nineteen other points. The second point only needs to be joined to eighteen points now. The third point needs to join seventeen points and so on. Number of lines, L = 19 + 18 + 17 + ......... + 1 = $\frac{1}{2} \times 19 \times [19 + 1]$ = $\huge190$ .

1st point will be conected to other 19 points with 19 lines.
2nd with 18 points with 18 lines (b/c it is already conected to 1st point) similarly.
3rd with 17 lines
4th with 16 lines
5th with 15 lines
.
.
.
.
.
20th with 0 line (b/c it is already coneced to all other points)
therefore total number of lines is:
19 + 18 + 17 + 16 + 15 + . . . . . . . . . . . . + 0 = 19(19+1)/2 = |190| -----> ANSWER