Consider a set with n elements, how many subsets (power set) does it have?
There is one way to calculate it. Each element has two choices, so that the total number of subsets will be 2^n
Another way, the cardinality of any subset must be >=0 and <= n. The number of subsets with cardinality k is C(n, k). So that the total number of subsets will
sum C(n, k)
2.
Consider polynomial (x+y)^n, expanding it
we have (x+y)^n = sum C(n, k) x^k y^{n-k}
set x = 1, y = 1
There are many ways more to prove it..