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How does...

well, Pascal's triangle is much faster (in my opinion) than the binomial expansion theorem. (unless of course, someone asks me for the 20th term of (2x + 5y)^55, and why they would want to know only 1 term, I don't know.)

I don't use Pascal's triangle daily, but probably once or twice a week, on average.
 
It also can be used in a lot of combinatorial mathematics, which comes into play any time you try to calculate probabilities for just about any nontrivial random event (such as playing poker, which I do on a fairly regular basis). 🙂
 
If you don't know how combinatorial math (in particular, choosing N items randomly from a set of M items) applies to poker and how it relates to Pascal's Triangle, I'm not sure I know where to start explaining. 🙂 Find a basic book on probability theory and start there.
 
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