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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 $A \to B \to C \to E \to F \to H \to I.$) Oct 12, 2023

#2
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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.

Oct 12, 2023

#1
-1

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.

Oct 12, 2023
#2
+1

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.

BuilderBoi Oct 12, 2023