How many ways (Solution)

20 ways.

Here are two good ways to solve it.

Label each vertex

The only way to get to a vertex is from below or from the left. If there are \(b\) ways to get to the vertex below and \(l\) ways to get to the vertex to the left, then there must be \((b + l)\) ways to get to this vertex.

To start, there’s only one way to get to the bottom left vertex. Work from there.

Combinatorics

If each move is a move one up or one to the right, then you will need to take exactly 6 moves to get from the bottom left to the top right. 3 of those moves, need to be moves up. The only thing left is to decide which 3 moves (out of your 6 moves) will be up moves. That’s \({6 \choose 3} = 20\)