Binomial Theorem

For integers r, n with 0 \< r < n, what is (ⁿ_r) + (ⁿ_r+1)?

(ⁿ_r) + (ⁿ_r+1) = (ⁿ⁺¹_r+1)

What is the binomial theorem?

For any integer n>/ 0 and any real numbers a, b, we have

(a + b)ⁿ = ∑ⁿ_i=0 (ⁿ_i) aⁱb^n-i

What does ∑ⁿ_i=0 (ⁿ_i) equal?

2ⁿ

What is a corollary about the sum of (ⁿ₀) - (ⁿ₁) + (ⁿ₂) - … +- (ⁿ_n) for n >/ 1?

(ⁿ₀) - (ⁿ₁) + (ⁿ₂) - … +- (ⁿ_n) = 0 where the +- at the end is +ve if n is even and -ve if n is odd

Let S be a set of size n >/ 1. How many subsets of even and odd size does S have?

2^n-1 even subsets and 2^n-1 odd subsets