When I think of bell curves, I tend to think more of Galton boards than a graphical representation of test score averages. Galton boards are those fun things with all the marbles in them that you flip over and most of them gather in the middle while a few go off to the sides. Big enough Galton boards can approximate a bell curve fairly accurately. Really interesting seeing how even though each ball follows an unpredictable path, altogether they form a shape that's quite predictable, if you have enough of them.
Anyhow, Galton boards have binomial coefficients in their mathematical calculations - like many things in probability do - and of course binomial coefficients have factorials. But what's interesting is factorials can be represented by Striling's approximation. This estimates n factorial to be pretty close to (root (2*pi*n))*(n/e)^n. Hopefully I typed all that out correctly, but I'm sure it's not hard to find online.
Well anyhow, that's how we see an e appear, if we trust formulas like that and feel comfortable looking up the derivations of those formulas later. Stirling's approximation is involved in the approximation of a factorial, and if you plug those approximations into the binomial coefficients in the formulas similar to Galton board type models, you get a formula called the De Moivre-Laplace Theorem.
The Galton board shows how a large accumulation of randomized left-right choices from a centralized point (which I guess we could think of as a mean) forms a bell curve, and the De Moivre-Laplace theorem gives the formula showing the e in that equation.
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