3 Ideas in Paths

Idea 1. Placing Numbers

1) Origin $=1$.

2) Number at each position $=$ the sum of the numbers at exactly the previous positions.

At each step, we can jump 1 or 2 units to the right or one unit upward.

# ways from $A$ to $B$

Idea 2. Permutation

At each step, we can jump 1 unit to the right or one unit upward.

# ways from $A$ to $B$

Each path corresponds to a permutation of 5 $R$'s and 3 $U$'s.

Idea 3. Levels

At each step, we can jump 1 unit to the right, one unit upward, or 1 unit to the left.

We can't pass through a point more than once.

# ways from $A$ to $B$

There are 4 levels (horizontal lines). If we only determine the places where we go to the next level, then the path will be determined.