like i said last night, i don't know any applications for Pascal's theorem, but i can give you two examples for pascal's triangle

pascal's triangle is particularly useful in expanding binomials, for example,

we know that

(x + 3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81

and

(x + 2)^6 = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64

using pascal's triangle to find the coefficients of all the terms

do you know how to do this?