Question

Solving for a variable in sets and combinations

Answer 1

n C k is just a shorthand for n!/(k! * (n-k)!). The binomial theorem states that when expanding (x + y)^(n), the x^(a)y^(b) term has coefficient n C a = n C b where a + b = n, so when x = y = 1, (x + y)^n = 2^(n). There’s 5 terms because you can choose between 0 through 4 for each exponent.

Answer 2

Think of this as the power set of a set of r elements.  

You can choose 0, 1, 2, .., elements in each subset.