# Number of ways to travel 20 blocks

Ying lives at location A and works at location B as indicated below. Note that Ying needs to travel at least 20 blocks to go to work, namely 10 block south and 10 block east. Find a formula for the number of ways Ying can get from home to work (under the condition that she travels only 20 blocks). This is the diagram: I know it isn't that hard but something about this just makes me stuck. I am thinking that the answer is 20C10 but I don't think it is correct, and how I would arrive to that.

• There are exactly 10 steps down, and 10 steps left. How many ways can you arrange them? Jul 1, 2013 at 23:28

You are correct, the answer is $\binom{20}{10}$. Think of it this way: to go from $A$ to $B$, you have to exactly $20$ blocks, and $10$ of them must be going south, and $10$ of them east. So you can think of a path as a sequence consisting of $20$ letters, $10$ of which are an $S$, and the other $10$ must be an $E$. The number of such sequences is $\binom{20}{10}$.