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:
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?