[Binomial theorem]: Apply the binomial theorem to prove that for any given natural number n, 1n, we have 1 = C(n,1)-C(n,2)+C(n,3)-C(n,4)+....+(-1)n+1C(n,n)? In other words, prove that 1 = -1 * 1in (-1)i C(n,i). Hint: think about the expansion of (-1+1)n. 5. [Pigeonhole principle]: Prove that for any set X of at least 5 distinct natural numbers there must exist two numbers m and n in X such that m>n and either m+n or m-n is divisible by 7