Alice and Bob have a farm in the form of a $20 \times 20$ grid, each square in the grid containing the same amount of fruit. They both live in the lower-right square. Luckily for them, Alice is a brilliant inventor, and has crafted a machine that can collect all of the fruit in the square in which it's located instantly. Unluckily for them, this machine only has the technology to move right or down, so they have to do the harvest by painstakingly carrying it to the top-left square (the machine doesn't work while carried) and making a single pass collecting as much fruit as they can.
This means that most of the field will be left unharvested, but they're lazy farmers and don't really care about that.
Bob, wanting to stump Alice, then asked "What if we carried it to the top-left square twice instead? In how many ways could we make the two trips from the top-left square to our home such that we collect the maximum total amount of fruit?"
However, he was unsuccessful, as Alice quickly entered a few numbers on her (scientific) calculator and showed him the answer.
What is the answer to Bob's question and how did Alice reach it? In other words, what is the number of non-intersecting pairs of paths from the top-left square to the bottom-right square?
Note: Two ways are considered distinct if in at least one of the two passes Alice and Bob take a different path home.
The condition "they don't meet in any point except for the endpoints" is a bit clumsy. Can we transform the problem into finding two paths that share no common points at all, even the endpoints?
Bob asked Alice how she reached her answer. I didn't hear the whole explanation, but saw she drawing this picture in her notebook: