A ruler with no mark on it can measure its own length $$AB$$. A ruler with only one mark on it can measure $$3$$ lengths $$AB,AC,BC$$. A ruler with two marks on it can measure $$6$$ lengths $$AB,AC,CD,DB,AD,CB$$. Then the number of lengths a ruler with $$4$$ marks on it can measure is

To determine any particular length we need to choose a set of two points from the given set of points on the ruler.

In the first case we had to choose 2 points out of 2 points given (ends of the ruler)  which is $$\binom{2}{2} =1$$ ways. 

In the second case we are given 3 points given (end points and one mark in between) out of which we must choose 2 points given which can be done in $$\binom{3}{2} =3$$ ways.

In the second case we are given 4 points (end points and two marks in between)  out of which we must choose 2 points given which can be done in $$\binom{4}{2} =6$$ ways.

Extending this to the last case we are given 6 points (end points and four marks in between) out of which we must choose 2 points given which can be done in $$\binom{6}{2} = \dfrac{6!}{4!2!} = 3\times 5 =15$$ ways.