Transcript for the Math Jam "AoPS Intro Counting & Probability and Intermediate Number Theory" on Feb 14.

Math Jam hosted by DPatrick (Dave Patrick ).

DPatrick (19:28:47)
Hello, and welcome to the first AoPS Winter/Spring 2007 Classes Math Jam!

DPatrick (19:28:54)
Before we get started I would like to take a moment to explain our Virtual Classroom to those who have not previously participated in a Math Jam or one of our online classes.

DPatrick (19:29:06)
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. This helps keep the class organized and on track. This also means that only well-written comments will be dropped into the classroom, so please take time writing responses that are complete and easy to read. Also, only moderators can enter into private chats with other people in the classroom.

DPatrick (19:29:34)
Note that it is not possible for the instructor to personally respond to every comment that you submit -- please do not take it personally if your comment is not posted or responded to! I will try to respond to all questions to the extent that I can.

DPatrick (19:29:54)
In addition, the virtual classroom is LaTeX enabled. LaTeX allows users to make nice equations and other math expressions. If you would like to learn how to write in LaTeX, click on the tab on the left side panel of our website and there is a tutorial and reference guide there.

DPatrick (19:30:05)
Using LaTeX in the virtual classroom is slightly different than using it on the message board or in a LaTeX editor. If anything you type up in a post uses LaTeX, then you must use a semicolon (;) to begin your post. For example, if you type

DPatrick (19:30:08)


DPatrick (19:30:14)
This message will look like this when posted in the classroom:

DPatrick (19:30:17)


DPatrick (19:30:31)
Just remember, if your post uses LaTeX, use the semicolon (;) to begin your post!

DPatrick (19:30:48)
One last thing: we recommend not to use a wireless connection while in the classroom. These have a tendency to cause disconnections. Please use a wired connection if possible.

DPatrick (19:31:01)
Today we will be discussing two of our upcoming winter/spring classes:
- Introduction to Counting & Probability (starts February 21 and runs for 12 weeks)
- Intermediate Number Theory Seminar (starts February 28 and runs for 8 weeks)

DPatrick (19:31:18)
There will be another Math Jam later this month to discuss to discuss the remainder of our winter/spring schedule (Intermediate Algebra and the Special AIME Problem Seminar; these classes both begin in March).

DPatrick (19:31:28)
All of our online classes take place in this Virtual Classroom.

DPatrick (19:31:48)
First, let's discuss Introduction to Counting & Probability.

DPatrick (19:31:56)
In the Introduction to Counting and Probability class, we cover basic and intermediate counting concepts, including casework, multiplication, permutations, combinations, Pascal's triangle, probability, combinatorial identities, and the Binomial Theorem.

DPatrick (19:32:10)
The main emphasis of this class is learning how to take an organized approach to counting, and understanding that nearly all of counting is learning when to use the basic arithmetic operations division, multiplication, addition, and subtraction (and of course why to use them when you use them).

DPatrick (19:32:29)
Students completing this course, who work most of the problems, should come out of the course knowing how to tackle any MATHCOUNTS Counting problems, most AMC 10 and AMC 12 counting problems, and even some AIME counting problems. The concepts in this course are also crucial to understanding computer science.

DPatrick (19:33:03)
We'll now take a look at a couple of sample problems from the course, which highlight some of the tactics we'll investigate in this class. (This will also give you an example of how the class will typically look.)

DPatrick (19:33:09)


DPatrick (19:33:16)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-86431359.gif

DPatrick (19:33:29)
Usually, after posting a problem, the instructor will also post a link to the problem, as I just did above. This way, the student can click on the link and keep the problem open in a separate window, so that he or she can follow along with the solution. (If clicking on the link doesn't do anything, try holding down the Ctrl key as you click. If it still doesn't work, you may need to change the settings on your pop-up blocker.)

DPatrick (19:33:46)
Does anybody know the answer to this problem?

pianoaddicted124 (19:33:52)
5040?

raklorap (19:33:57)
7 factorial

DPatrick (19:34:20)
Right -- though in an actual class, I would never just post the answer like this. Instead, the class would work through it, as follows:

DPatrick (19:34:45)
This is a straightforward application of multiplication: there are 7 students who could sit in the first seat. For each of these choices we make for the first seat, there are six ways to choose a student for the next seat, so there are 7 x 6 ways to seat the first two students.

DPatrick (19:35:09)
Continuing in this vein, for each of these 7 x 6 ways to seat the first two students, there are 5 ways to pick a student for the third seat. Thus, there are 7 x 6 x 5 ways to seat the first three students.

We keep going like this: there are 4 ways to seat the fourth student, 3 ways to seat the fifth, 2 ways to seat the sixth, and one way to seat the last student. This gives us 7 x 6 x 5 x 4 x 3 x 2 x 1 ways to seat all the students.

DPatrick (19:35:43)
We run into products like 7 x 6 x 5 x 4 x 3 x 2 x 1 so much in mathematics that we have a symbol and a name for it. We write 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! and we call this 'seven factorial'.

DPatrick (19:36:01)
That problem was pretty simple. Let's put a wrinkle in it. Suppose we must have a girl in the first chair and a girl in the last chair. Then how many seatings are there?

DPatrick (19:36:02)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-172749551.gif

galbatorix (19:36:29)
5 factorial?

speight (19:36:30)
5 factorial

Chocoboom (19:36:56)
5 factorial because there are 5 seats left

DPatrick (19:37:19)
Except that we have choices for which girls go in the end seats.

DPatrick (19:37:27)
How do we account for that?

katherine oconnor (19:37:29)
1440?

DPatrick (19:38:06)
That's correct -- how did we get that?

cat810930 (19:38:06)
4*3*5*4*3*2*1=1440

DPatrick (19:38:31)
Right. Here's the explanation:

DPatrick (19:38:41)
We can think to ourselves 'How would we seat the kids according to these restrictions if we had to make up a seating ourselves?' Our answer is: we'd seat the girls at the ends first, so we make sure we satisfy that restriction.

DPatrick (19:38:53)
As before, there are 4 ways to seat a girl in the first seat. Next we seat a second girl in the last seat - there are 3 girls left, so there are 3 choices. Now we have our restriction taken care of. We can then seat the rest of the students as before. There are 5 students left to choose one for the second chair, then 4 students for the third chair, and so on.

DPatrick (19:39:21)
Thus, we have 4 x 3 ways to seat girls at each end, and for each of these seatings we have 5 x 4 x 3 x 2 x 1 ways to seat the rest of the students, for a total of:

4 x 3 x 5 x 4 x 3 x 2 x 1 = 1440

DPatrick (19:39:31)
This example brings up two important counting concepts.

DPatrick (19:39:47)
First, when dealing with a counting problem that has restrictions, it often pays to think about how you would create one possible arrangement yourself. Here, we realize that if we seated the students ourselves, we'd start with the girls on the ends. This brings us to our second important counting concept:

DPatrick (19:39:52)
When dealing with restrictions, it usually helps to deal with the restrictions first. Here, we took care of the girls on the ends first since that was our restriction.

DPatrick (19:40:05)
Let's put a different twist on this problem:

DPatrick (19:40:11)
We still have 7 students to seat in a row, but two of them, Ali and Brianna, refuse to sit next to each other. In how many ways can we seat the students now?

DPatrick (19:40:14)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-131867599.gif

DPatrick (19:40:59)
What's wrong with this solution:

There are 7 ways to seat Ali. We deal with the restriction first and realize that we can't seat Brianna in either of the seats next to Ali. Hence, Brianna has 4 choices. Then the next student has 5 choices, the one after that has 4 choices, and so on.

What's wrong?

katherine oconnor (19:41:34)
Ali might sit on an end.

DPatrick (19:41:46)
Right. The problem here is that there are not always 2 seats next to Ali - sometimes he may be put at the end. Hence, sometimes Brianna will have 5 choices for her seat.

DPatrick (19:42:00)
We could deal with this by using casework (and we'll discuss very important casework strategies in the course - these tricky casework problems are often the difference in proceeding to the next level in MATHCOUNTS/AMC), but there is a slicker approach.

Ubemaya (19:40:55)
copmlementary counting

DPatrick (19:42:35)
Right. We can count the number of ways in which Ali and Brenda are seated next to each other, and subtract this from the 7! total ways in which the students can be seated.

DPatrick (19:42:50)
In how many ways can we seat the 7 students if Ali and Brianna are together?

redcomet46 (19:43:18)
Well, 2*6!

DPatrick (19:43:36)
Right.

DPatrick (19:43:47)
We can pretend Ali and Brianna are one person: AliBrianna. Then, we have 6 students and we have no restrictions. Thus, we have 6! ways to seat these students.

DPatrick (19:43:54)
But Ali and Brianna are not the same person. They could be AliBrianna or BriannaAli. Thus, for each of our 6! seatings, there are 2 orders in which we can seat Ali and Brianna in their slot. Hence, there are 2 x 6! ways to seat the students such that Ali and Brianna are together.

DPatrick (19:44:16)
So the answer to the original problem is 7! - (2*6!) = 3600.

DPatrick (19:44:29)
This example brings up a couple more important tactics.

First, when it looks hard to count something directly, try counting the opposite of what you're asked for. We call this approach complementary counting, since 'complement' in dealing with groups of objects in mathematics roughly means 'opposite'. I also call this 'counting what you don't want'.

Second, when your restriction is that some of your items must remain together when putting them in a row, a useful tactic is to consider the items all together as a single item, as we did AliBrianna above. Then you separately consider how many ways you can order the items within the group.

raklorap (19:44:31)
why is it 6!

DPatrick (19:45:14)
Because when we consider Ali & Brianna as "one person", we then have 6 "people" to seat: the 5 other students plus AliBrianna.

DPatrick (19:45:30)
(In other words, imagine that they're glued at the hip.)

DPatrick (19:45:48)
These basic examples show why it is pointless to memorize your way through counting - I can ask zillions of variations of the above questions. Instead of memorizing your way through each variation, you should learn when to add, when to subtract, when to multiply, and when to divide. Since you already know how to perform these operations, once you know when to do them, you know how to count!

DPatrick (19:46:03)
The first of these problems was considerably easier than most of the problems we will do in the course. The second and third are closer to the middle in difficulty, though they are still a little easier than the average problem.

DPatrick (19:46:17)
In general in the course we will go through the ideas much more gradually than we did here - each idea will be explored with gradually more difficult examples. Thus, the pace at which new ideas are introduced is slower than we did here (in which we introduced 4 general tactics in two problems!)

DPatrick (19:46:37)
You can find more questions like those that we cover in the course by checking out the Post Test for the course here:

http://www.artofproblemsolving.com/Classes/IntermCounting/PostTest.pdf

DPatrick (19:46:54)
This course will be a bit different than other Art of Problem Solving courses in that we are using a textbook in conjunction with the course: our own Introduction to Counting & Probability book. The material covered in the textbook is roughly equivalent to the material covered in the course. You can see the table of contents and some excerpts from the book here:

http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=3

DPatrick (19:47:10)
The book is not required; however, students who have the book will be able to read additional material that complements the lectures, and will have access to a large number of practice problems at varying levels of difficulty. We are recommending that students read the corresponding chapter(s) in the book before each lecture, and attempt some of that chapter's Review and Challenge Problems after each lecture. We also expect to spend some class time answering students' questions about problems from the textbook.

DPatrick (19:47:30)
The homework for the class consists of weekly problems that will be posted to the class message board -- for these problem, you do not turn your solutions in, however you may post them to the message board if you like. The class also has two longer problem sets -- a midterm and a final -- for which you should write up your full solutions and submit them. These solutions will be read, and you will receive detailed feedback.

DPatrick (19:47:42)
Before we move on to the next class, are there any questions about this class or the textbook?

Shivar (19:48:10)
How long does the class last?

DPatrick (19:48:31)
Each class is 90 minutes long -- from 7:30-9:00 PM Eastern Time -- and the course lasts for 12 weeks.

cjaj2 (19:48:18)
what Grade is this course

DPatrick (19:48:51)
For most students, this course is appropriate for grades 7-10.

DPatrick (19:49:23)
It does not require any algebra beyond the very basics (like solving 6x + 3 = 21 or something like that).

bhavana (19:49:14)
is this class everyday or every wednesday

DPatrick (19:49:42)
Every Wednesday, starting one week from today.

raklorap (19:49:13)
Just wondering, do we pay for these classes? And, if so how much are they?

DPatrick (19:50:01)
The tuition for this class is $150 without the textbook, or $180 with the textbook.

pianoaddicted124 (19:49:18)
So would it be applicable both to highschool math and Mathcounts?

DPatrick (19:50:31)
Yes -- I would say that most of the problems are at the Mathcounts or AMC 10/12 level.

Shivar (19:49:31)
Is the course self-paced?

DPatrick (19:51:04)
It depends by what you mean by "self-paced". The classes are live, real-time (like what we're doing right now).

DPatrick (19:51:19)
But if you miss a class, you can read the transcript on your own time.

cjaj2 (19:49:49)
Any exams, passing grade, etc?

DPatrick (19:51:30)
There are no grades.

DPatrick (19:51:37)
There are weekly problems that you can do on your own.

DPatrick (19:52:03)
There are 2 longer problem sets -- one midterm and one final -- that you write up and turn in. We'll read those and give you comments and feedback.

Shivar (19:50:37)
Approximately how many hours would we have to dedicate to the class outside of the class itself?

DPatrick (19:52:34)
If you're going to read along in the textbook along with the class, then 1-2 hours/week for the reading.

DPatrick (19:52:54)
There are 10 weekly problems posted after each class, so 1-2 hours to do those.

DPatrick (19:53:29)
The longer problems sets are 15-20 problems each, and you'll be writing up full solutions, but they're spread over 6 weeks, so maybe 3-4 hours total for those (again spread over 6 weeks).

pianoaddicted124 (19:51:56)
Are those required to complete (like for homework?)

xharryyc_2 (19:51:59)
so the homework is going 2 be on the website?

DPatrick (19:54:00)
Everything for the class will be posted on the website. Nothing is "required" in the sense that we don't give grades.

pianoaddicted124 (19:53:22)
Do the problems get harder at the same rate as the class's math difficulty?

DPatrick (19:54:26)
Yes and no. The material naturally gets more difficult as the class progresses, but every week, there will be some easy problems and some hard ones.

xharryyc_2 (19:53:01)
Are all the problems that we are going to solve in the textbook?

DPatrick (19:55:07)
No -- in fact, the textbook is not required. Some of the problems that are posted on the website will be taken from the textbook, but some will not.

xharryyc_2 (19:54:57)
Will we have to eventually use LaTeX to answer problems, or can we just type them normally?

DPatrick (19:55:37)
LaTeX is not required. You don't even have to type them -- you can handwrite your solutions and mail them in.

nitish (19:55:33)
to whom do we pay the money?

DPatrick (19:56:13)
You can enroll by going to our website and clicking on "Enroll" in the "Online School" tab on the left side of the page.

DPatrick (19:56:43)
We'll have more time for questions at the end -- let me now discuss the Intermediate Number Theory Seminar.

DPatrick (19:56:53)
The Intermediate Number Theory Seminar is an 8 week course for students who have both a strong foundation in basic number theory and solid algebraic skills. It is recommended that students take the Introductory Number Theory course and Intermediate Algebra course before enrolling, or be confident with the material covered in those classes.

DPatrick (19:57:12)
Topics covered in the Intermediate Number Theory Seminar include algebraic methods of problem solving in number theory, base number problems involving algebra, counting, and qualitative problem solving techniques, divisibility and divisor problems involving algebra, Diophantine equations, modular arithmetic with an emphasis on algebraic applications, perfect squares, an introduction to Pell's equations, Fermat's Little Theorem, Euler's Phi Function, and Euler's Theorem.

DPatrick (19:57:51)
As you can probably tell, this is a much more difficult class than the counting class that we just discussed. This class is intended for more experienced students.

DPatrick (19:58:00)
We will now work a few problems from different days in the class. We will start with an easier class problem and move into harder ones. Unfortunately, we cannot sample some of the most important topics covered in class because it would take too much time developing the background to discuss them here.

DPatrick (19:58:07)


DPatrick (19:58:12)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-72995614.gif

DPatrick (19:58:31)
How can we approach this problem?

daermon (19:58:26)
factor the quadratic?

DPatrick (19:58:42)


DPatrick (19:58:50)
How does this factorization help us?

daermon (19:58:59)
we know that its divisible by 2

DPatrick (19:59:49)
So we only need to worry when f(s) is a multiple of 3.

DPatrick (20:00:06)
When will f(s) be a multiple of 3?

Ubemaya (20:00:39)
when x is not a multiple of 3

DPatrick (20:00:58)
Right. f(s) = (s+1)(s+2) will be a multiple of 3 when either s + 1 or s + 2 is a multiple of 3. This is the same thing as saying that f(s) will be a multiple of 3 when s is NOT a multiple of 3.

DPatrick (20:01:23)
So we now count the members of S that aren't multiples of 3 and we count a total of 17 of them. Thus 17 is the answer.

DPatrick (20:01:44)
The key to this problem was factoring the function. Once we did that, the actual divisibility arguments were relatively straightforward.

DPatrick (20:02:16)
As I said, this is a more difficult course. If you found the previous problem very difficult, then you are probably not ready for this class, as this is a relatively easy problem in the course.

DPatrick (20:02:30)
Let's try one more:

DPatrick (20:02:35)


DPatrick (20:02:37)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-260159023.gif

DPatrick (20:03:12)
Where do we start?

bhavana (20:02:56)
5

DPatrick (20:03:41)
Good -- 5 is an example. 5 in base 2 is 101, and 10 in base 3 is also 101.

DPatrick (20:03:53)
It's often good to have examples to experiment with.

DPatrick (20:04:29)
Any other examples of the top of anybody's head?

pianoaddicted124 (20:05:13)
3

.cpp (20:05:22)
0

DPatrick (20:06:09)
Well, 3 doesn't work, but 0 does.

DPatrick (20:06:20)
3 is 11 in base 2, but 6 is 20 in base 3.

7h3.D3m0n.117 (20:06:40)
6?

DPatrick (20:07:25)
6 is 110 in base 2, and 12 is 110 in base 3.

DPatrick (20:07:33)
These are acutally all of them.

DPatrick (20:07:51)
Did these examples give any clue how to prove it?

nitish (20:08:07)
how do you do it without guessing?

DPatrick (20:08:24)
Let's try to figure that out.

DPatrick (20:09:46)
sorry, my computer flaked out for a moment...

DPatrick (20:09:52)


DPatrick (20:10:13)


DPatrick (20:10:31)


DPatrick (20:10:48)
We can now apply what I like to call the Fundamental Principle of Algebraic Problem Solving -- find two ways to express the same thing and set them equal!

DPatrick (20:10:53)


DPatrick (20:11:13)
Now what?

daermon (20:11:10)
hmm now subtract from both sides?

DPatrick (20:11:36)


DPatrick (20:11:57)
Now, what do we notice about those coefficients?

redcomet46 (20:12:05)
THey become negative

Ubemaya (20:12:05)
all are positive except one

DPatrick (20:12:29)
Right, all the coefficients after the a_0 and a_1 terms are negative!

DPatrick (20:12:38)


DPatrick (20:13:09)


DPatrick (20:13:39)
So this gives us a bound for k (the length of the number)!

DPatrick (20:13:57)


DPatrick (20:14:21)
So all the solutions are between 0 and 7. We can simply check them all and we see than 0,5,6 (that we found earlier) are all of them.

DPatrick (20:14:32)
So the answer to the original problem is that there are 3 solutions.

DPatrick (20:14:38)
The key to this problem was using a sequence of variables to represent the digits of n so that we could use these variables to compare the two different ways we had to express the value of n. Once we did that, we could fall back on our algebraic skills using equations and inequalities.

raklorap (20:14:39)
so you still have to use guess and check

DPatrick (20:14:50)
At the very end, yes.

DPatrick (20:15:22)
So as you've seen, this class does require a certain level of familiarity with both algebra and basic number theory (such as base numbers).

DPatrick (20:15:50)
There's a pre-test for this class, if you'd like to see whether you're ready for this class:

DPatrick (20:15:56)
http://www.artofproblemsolving.com/Classes/IntermNT/PreTest.pdf

DPatrick (20:16:21)
(It's also available by going to the "Online School" page on the website and clicking on "Enroll")

bhavana (20:16:23)
is there a pre-test for the other class

DPatrick (20:16:32)
Yes, there is:

DPatrick (20:16:37)
http://www.artofproblemsolving.com/Classes/IntroCounting/PreTest.pdf

DPatrick (20:16:55)
It's mainly just a test of your algebra basics.

raklorap (20:16:55)
do you normally use calculators on the pre-tests

DPatrick (20:17:17)
It's up to you.

pianoaddicted124 (20:17:19)
Is it timed?

DPatrick (20:17:40)
No, and you don't have to take it. It's just for you to gauge your own level of preparedness for the class.

nitish (20:17:32)
can we get this transcript because it will not highlight to copy?

DPatrick (20:18:01)
The transcript will be posted on the website after the session is over.

KingRoy (20:17:35)
How do I turn in the assignment?

DPatrick (20:18:24)
Weekly problems are not turned in -- you just do them on your own, and you can post your solutions on the message board if you like (but it's not required).

DPatrick (20:18:35)
The longer problem sets can be submitted by mail, email, or fax.

bhavana (20:17:45)
what grade levels is the second class for

DPatrick (20:18:56)
It's for more experienced problem solvers. You need to have taken algebra.

.cpp (20:18:46)
Is there a certificate for this course?

DPatrick (20:19:26)
Yes. we'll issue a certificate of completion upon request.

dkang00 (20:18:05)
So, what's considered a good score?

DPatrick (20:19:36)
A good score on what?

raklorap (20:19:35)
what would a certificate do

DPatrick (20:19:43)
Hang on your wall. :)

dkang00 (20:19:41)
the pretest

DPatrick (20:20:04)
You should be able to master the pretest. You have to decide for yourself what that is.

bhavana (20:19:51)
do we get credits for this course

DPatrick (20:20:39)
Not usually. We can occasionally arrange it with your school.

cjaj2 (20:20:03)
The first class is for Gr. 7 to 12 but you also said it's more at the 10/12 level. Will a Gr. 7 gifted in math be able to manage?

DPatrick (20:21:10)
If you can do the pretest, then you're ready for the course. That's the best way to judge.

bhavana (20:21:13)
what is AMC

DPatrick (20:21:24)
American Mathematics Competitions. www.unl.edu/amc

DPatrick (20:21:51)
There are three of them: AMC 8, AMC 10, AMC 12 designed for grades 8, 10, 12.

.cpp (20:21:49)
How can I request a certificate (or is it automatic?)

DPatrick (20:22:05)
It's not automatic -- just email the instructor at the end of the course.

KingRoy (20:21:56)
How many hours do you recommend to put on introduction p&c weekly in order to keep up the course?

DPatrick (20:22:29)
1-2 reading and 1-2 working the weekly problems. Then 3-4 hours spread over 6 weeks to do the problem sets.

DPatrick (20:22:51)
Any other questions?

daermon (20:23:02)
only 3-4 hours? i think they take longer :|

DPatrick (20:23:22)
Actually, probably so. But again, it's spread out over 6 weeks.

.cpp (20:22:59)
I want to make sure - do I have to e-mail my instructor for a certificate with every AoPS course?

DPatrick (20:23:33)
Yes.

Chocoboom_4 (20:23:27)
How much of the material in these courses are in the 2 AoPS textbooks?

DPatrick (20:24:22)
Much of the Intro Counting class is in the Volume 1 & 2 books, although in not as much detail. As I mentioned earlier, the class is essentially the same material as in the Intro Counting & Probability textbook.

DPatrick (20:24:42)
The Intermediate Number Theory class has some material in Volume 2, and some beyond.

choccharlie (20:24:36)
Do all of the classes cost the same?

DPatrick (20:25:06)
No. The Intermediate NT class is $98.

DPatrick (20:25:22)
(It's only 8 weeks and has no problem sets to be submitted for feedback.)

KingRoy (20:24:07)
If I have any question of the course or would like to discuss the solution of the problem , how do I do?

DPatrick (20:25:56)
There will be a message board set up for each class. That's where you can discuss the class and the problem.

raklorap (20:24:12)
How much is a textbook?

DPatrick (20:26:27)
The Intro C&P textbook is $39 (with solutions manual) or $33 (without). It's $30 if you enroll in the class.

ronviola (20:24:14)
are you going to have more books? like your book on probability is an introduction so will there be other ones?

DPatrick (20:26:48)
Yes, there will be Intermediate-level books as well. The Intermediate Counting & Probability book should come out this summer.

DPatrick (20:26:54)
(maybe sooner)

daermon (20:25:03)
since the interm number theory is a seminar, how much time commitment would it be>

DPatrick (20:27:31)
Probably 2-4 hours/week reviewing the lectures and working on the weekly problems. (Of course, you might spend less or more, it depends on the student and how much work they want to put into it.)

DPatrick (20:28:01)
If you have further questions, you can always post them in the "Classes Information" forum on the message board.

DPatrick (20:28:10)
Thanks for coming tonight!