Real world applications of Binomial Coefficients

If you're only looking for an overview of the fields using binomial coefficients, I suggest this short article.

But, obviously, this is just an overview, there is a huge amount of fields relying on the binomial coefficients, because they are way easier to work with, and because they appear pretty often.

To quote, the article, we can find the binomial coefficients in Albert Einstein's theories (which have obviously a lot of real-life applications), in protocols for the web, in architecture, finance, and a lot more.

And the binomial coefficients are, indeed, as you said, a major pillar of probabilities, which are extremely important in our world.