How many ways can I go from 1 to 10 in the following diagram?

Question

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:

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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?

Answer 1

You can think the problem as start with the $10$ and follow the numbers in order until you reach $1$.

Then there is two ways for each step: up or left. Then there are $2^{10-1}=2^9=512$ ways. Done!

Answer 2

Each square shows the number of (legal) paths to that square.

So the number of possible paths to the bottom right square is 512.