I'm working on a question from Harvard's Stat 110 course.
The question is the following:
3(a) How many paths are there from the point (0, 0) to the point (110, 111) in the plane such that each step either consists of going one unit up or one unit to the right?
I understand how to answer this: $${221 \choose 110} \equiv {221 \choose 111}$$
And I understand the reasoning:
You can place the movements up and right in a sequence. Selecting the positions of the movements right in this sequence automatically specifies the movements up too. (Likewise for the converse).
This makes sense to me intuitively. But I feel like there's a deeper explanation, which I can't find anywhere online.
What is the deeper explanation? Why do Binomial Coefficients work for problems like this?
You have a distance of 221 segments to cross,110 of which will be steps to the right, and 111 will be steps vertically.
We could put the vertical segments and the horizontal segments onto scrabble tiles and how many ways are there to select scrabble tiles…
This is the exact same process of
$(a+b)^{221} = (a+b)(a+b)\cdots(a+b)$
The coefficient of the $a^{110}b^{111}$ term counts the number of ways to pick the right number of $a$’s vs. $b$’s from the available buckets.