Chessboard possible ways from A to B

Question

How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right.

Lets say I have to travel from
x1 = 1, y1 = 1
to
x2 = 2, y2 = 2

Then there are 2 possible ways.

But what about
x1 = 1, y1 = 1
to
x2 = 8, y2 = 8

Is there some sort of a formula? Something with vectors?

Answer 1

We need to make in any case

• $8$ step on the right
• $8$ step up

therefore the number of possible ways is

$$\binom{16}{8}=\frac{16!}{(8!)^2}=\frac{16\cdot 15\cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10\cdot 9}{8\cdot 7\cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot2\cdot 1}=12\,870$$

since once we have chosen the 8 step on the right (or up) the path is completely determined.