Post-Exam Re-Hash

So... On Thursday, I had my Discrete midterm. Here's some ruminations on the exam:

1) There was a proof that looked like this: Use the binomial theorem to prove that the sum of all nCr(n,i)=2^n. Problem is, I couldn't recall until about halfway through the exam what the binomial theorem states. Here's what I put:
Pf: The Binomial Theorem states that (a+b)^n=nCr(n,n)*(a^n)*(b^n-n)+...+nCr(n,0)*(a^n-n)*(b^n). Then, let a,b=1. So, (1+1)^n=nCr(n,n)*(1^n)*(1^n-n)+...+nCr(n,0)*(1^n-n)*(1^n)=nCr(n,n)+...+nCr(n,0)=2^n. QED.
I know, pretty facile, right? But it's not wrong, just simple! Still, I'm not sure if I wrote it formally enough.

2) There was also this problem, which took me a LONG time: Let n1+n2+n3+n4=23, where n1>=2, n2>=3, n3>=4, n4>=5. How many solutions are there, where ni is an integer?
Solution: Due to the quantifiers, there are 2+3+4+5=14 numbers already accounted for. Also, there are 4 integers ni, so there are 3 dividers. So, the number of solutions is nCr(23-14+4-1,23-14)=(12,9).

3) Finally, there was this little stumper: How many (distinct) ways are there to rearrange the letters of TALLAHASSEE, such that the two L's are together and the two E's are NOT together?
Solution: First, let's find the number of ways to rearrange the letters with the two L's together. Consider the two L's to be a single letter. Then, there are 10!/3!2!2! arrangements. However, this includes the ways in which the two E's are together. So, let's find the ways in which the two L's are together AND the two E's are together, and subtract that number from the ways in which the two L's are together. This number is, then, 10!/3!2!2!-9!/3!2!.
I'm not sure I did that one right, but I feel pretty good about it.

And today I had my Group Theory midterm. I think I did pretty well on that one, too, but I did have a little trouble proving the onto-ness of the isomorphism P(x)=x^5. There were a few other isomorphism problems that I might have not done totally correctly, but only because they involved external direct products, and I'm still a little shaky on those. I'll post my exam results. Mostly to show off, of course, but, since no one actually reads this besides me, it's also a great way for me to... I don't know. Ok, ok, it's mostly for me to show off. Deal with it.