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.
- (5 points) Verify that the formula $latex \binom{n}{r}$ 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:
$latex (x+y)^n = C(n,0)x^n y^0 + C(n,1) x^{n-1}y^1 + C(n,2) x^{n-2}y^2 + \ldots + C(n,n) x^0 y^n$
So for example, $latex (x+y)^3 = x^3y^0 + 3x^2y + 3xy^2 + y^3$.
- (2.5 points) Use the Binomial Theorem to show that the number of all subsets of the set {1,2,…,n} is $latex 2^n$. (Hint: problem number 5 from the practice problems for Pascal’s Triangle may be helpful.)
- (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.)