Extra Credit!

I know all of you are thinking: Man! I’m totally going to miss finite math and all of these counting problems over spring break!

Well never fear! Here are some additional practice problems. Hand in these problems 5 minutes before class on Tuesday March 15 (or via email) to receive up to 10 extra credit points toward your quiz grade.

  1. (5 points) Verify that the formula \binom{n}{r} satisfies Pascal’s Indentity. In your solution, do not plug in numbers for n and r. (Hint: A small example may be helpful–check out your lecture notes from Tuesday March 1.)

For the next two problems you will use the Binomial Theorem which says:

(x+y)^n = C(n,0)x^n y^0 + C(n,1) x^{n-1}y^1 + C(n,2) x^{n-2}y^2 + \ldots + C(n,n) x^0 y^n

So for example, (x+y)^3 = x^3y^0 + 3x^2y + 3xy^2 + y^3.

  1. (2.5 points) Use the Binomial Theorem to show that the number of all subsets of the set {1,2,…,n} is 2^n. (Hint: problem number 5 from the practice problems for Pascal’s Triangle may be helpful.)
  2. (2.5 points) Use the Binomial Theorem to show that the alternating sum of each row of Pascal’s Triangle is zero. (Hint: problem number 6 from the practice problems for Pascal’s Triangle may be helpful.)