+1791 Find the number of paths from $A$ to $I,$ if each step must be in a right-ward direction. (For example, one possible path is $A \to B \to C \to E \to F \to H \to I.$)
+2666 Notice that there is only 1 path to get to B from A.
Then there is 1 way to get to both C and D from B.
Because you have to get to E from either C or D, there are 1 + 1 = 2 ways to get to E.
Then there are 2 ways to get to F and G because you have to go through E.
So then there are 2 + 2 = 4 ways to get to H.
And there is only 1 way to get to I from H, so there are a total of \(\color{brown}\boxed{4}\) paths from A to I.
+618 There are five possible paths from A to I, if each step must be in a right-ward direction:
A→B→C→E→F→H→I
A→B→D→E→F→H→I
A→B→D→G→H→I
A→C→D→F→H→I
A→C→D→G→H→I
This is because there are two choices for the first step (to B or C), two choices for the second step (to D or E), and two choices for the third step (to F or G). The remaining steps are then forced.
+2666 Notice that there is only 1 path to get to B from A.
Then there is 1 way to get to both C and D from B.
Because you have to get to E from either C or D, there are 1 + 1 = 2 ways to get to E.
Then there are 2 ways to get to F and G because you have to go through E.
So then there are 2 + 2 = 4 ways to get to H.
And there is only 1 way to get to I from H, so there are a total of \(\color{brown}\boxed{4}\) paths from A to I.
