TITLE: How many ways can I go from 1 to 10 in the following diagram?
QUESTION [9 upvotes]: This is a basic question in combinatorics with a little trick.
Consider the following triangular array of numbers:

How many paths from a 1 on the diagonal to the 10 in the lower right where we only step to the right or down are there? For example, here is a legal path:


Batominovski's edit:
Attempt:  It looks like each path is associated to a sequence of down steps ($D$) and right steps ($R$) of length $9$.  The example above corresponds to $DRDDDRRDD$.  Can it be any sequence?  What is a correct answer?

REPLY [30 votes]: Each square shows the number of (legal) paths to that square.
So the number of possible paths to the bottom right square is 512.