From ecroot@math.gatech.edu Sun Jun 6 15:13:23 2010 Date: Sun, 6 Jun 2010 15:13:23 -0400 (EDT) From: Ernie Croot To: Henry Mei Subject: Re: MATH 3012 grading Dear Henry, My apologies for not getting you the answer sheet sooner. Grade the following problems for accuracy out of 20 points each (for a total of 60 points), and grade the rest for effort: Problems 3, 4, and 5. Make sure that they show their work -- I am more interested in how they got to the answer than I am their actual answer. Please get them back to me by Tuesday, June 15 if at all possible. Here are the solutions: 3. To solve that problem, we must count the number of solutions where the x_i's sum to at most 9 and then subtract the solutions where they sum to at most 20. How many sum to at most 9? Well, first we add a slack variable x_5, to convert the problem into x_1 + ... + x_5 = 9, where x_1, x_2, x_5 >= 0, and where x_3,x_4 >= 1. Then, we make a change-of-variable, setting y_3 = x_3 - 1, y_4 = x_4 - 1, so that the equation becomes x_1 + x_2 + (y_3 + 1) + (y_4 + 1) + x_5 = 9, where x_1, x_2, y_3, y_4, x_5 are all >= 0. Rewriting this equation, we get x_1 + x_2 + y_3 + y_4 + x_5 = 7, which we know has (11 choose 4). When we do the same thing for solutions to x_1 + x_2 + x_3 + x_4 <= 20, we get (22 choose 4). So, the number of solutions we seek is (22 choose 4) - (11 choose 4) = 6985. 4. To determine the coefficient of a^8 b^3 in (2a^2 b - 7a + 3)^10, we note from the multinomial theorem (x+y+z)^10 = .... + (10 choose u,v,w) x^u y^v z^w + ... that we are working with x = 2a^2 b, y = -7a, z = 3, and u = 3, v = 2, w = 5. So, the coefficient is (10! / 3! 2! 5!) * 2^3 * (-7)^2 * 3^5 = 240045120. 5. We first count the number of permutations, and then subtract the number HAVING consecutive B's. The total number of permutations is 10! / 2! 3! 2! 3! = , and the number HAVING three consecutive B's is just the number of permutations of AABCCDDD (For each permutation of this string, we replace the B symbol with triple-B symbol BBB.) This number is 8! / 2! 1! 2! 3! . The difference between these counts is 23520. IT MIGHT BE A GOOD IDEA TO CHECK MY ANWERS -- I WROTE THIS UP RATHER HURRIDLY, AND AM PRONE TO MAKING NUMERICAL ERRORS. Best wishes, ERnie On Fri, 4 Jun 2010, Henry Mei wrote: > Dr. Croot, > > I have received the stack of HW #1s from Sharon McDowell, but I could not > find the solutions sheet you said I would be provided. It'd be great if you > could send it to me, so I can grade the papers over the weekend. > > Thanks, > Henry > > On Wed, Jun 2, 2010 at 1:16 PM, Henry Mei wrote: > >> Sorry, I just realized I have an eye/contact lens appointment at 8. I'll >> try to make it to your office by 9, but just know that I may and probably >> will be a bit late. >> >> Henry >> >> >> On Wed, Jun 2, 2010 at 1:05 PM, Ernie Croot wrote: >> >>> >>> >>> Ok, how about meeting with me at 9:00 tomorrow? >>> >>> And yes, I will provide solutions. >>> >>> Best, >>> >>> Ernie >>> >>> >>> On Wed, 2 Jun 2010, Henry Mei wrote: >>> >>> Prof. Croot, >>>> >>>> Unfortunately, I have to TA for ECE 2031 Digital Design Lab from 12-1:30 >>>> ( >>>> class later at 2:40 that afternoon) and I have class from 10:40-11:50 in >>>> the >>>> morning. I'd be happy to meet with you before your class to discuss the >>>> HW >>>> and then collect the HW sometime later that day... or just meet you in >>>> the >>>> afternoon. Let me know. >>>> >>>> With respect to the HW, will I be provided solutions, or will I need to >>>> create my own key? >>>> >>>> Henry >>>> >>>> On Wed, Jun 2, 2010 at 12:21 AM, Ernie Croot >>>> wrote: >>>> >>>> >>>>> >>>>> Dear Amanda and Henry, >>>>> >>>>> I will collect the first HW this Thursday. Henry, could you perhaps >>>>> meet with me after class on Thursday? >>>>> >>>>> The HW and syllabus are on my website at >>>>> >>>>> www.math.gatech.edu/~ecroot/ >>>>> >>>>> Click on Math 3012. >>>>> >>>>> Best, >>>>> >>>>> Ernie >>>>> >>>>> >>>>> On Tue, 1 Jun 2010, apascoe3@math.gatech.edu wrote: >>>>> >>>>> Hi Professor Croot, >>>>> >>>>>> >>>>>> I just wanted to touch base with you about grading. Klara has assigned >>>>>> Henry Mei to grade for the class; I'm still not sure if this is before >>>>>> I >>>>>> start teaching or after, but I emailed her to ask. In any event, I was >>>>>> wondering if you have assigned any homework that needs to be graded. >>>>>> Also, >>>>>> have you given out a syllabus? >>>>>> >>>>>> Thanks, >>>>>> Amanda >>>>>> >>>>>> >>>>>> >>>>>> >>>> >> >