Summary:

We took the midterm exam today, covering Chapters 1 to 6 (Countings). Congratulations to Annabelle, Jason, and Susan for being Top 3 this time! (Max is absent because of his contest). Keep up the good work!

If you feel upset about the score, do not give it up. What more important is to learn from the mistakes, and find the area of improvement. Look ahead. I am looking forward to your progress in the following months!

Here is the link to the answer key.

Here are some pictures of the board-writings in our second period (解题课), to help you make corrections.

The first picture is for Question 15. Thanks a lot to Annabelle for explaining it very clear to the class!!

The second picture is for Bonus Question 2. The star represents women, and the dot represents men. Please feel free to reach out if you have any questions.

Homework:

1. Make corrections to the test (Use a different color).

2. Preview Chapter 7: Introduction to Probability

See you next week!

Welcome back, everyone!

Lesson Review_Chapter 6: Some Harder Counting Problems

Tips:

1. Plug in some numbers for variables, solve, and look for a pattern.

2. Try to solve a simpler version of the same problem.

3. Be flexible. Try different approaches.

Homework:

1. Page 109, Ex 6.8 to 6.12 (only 5 problems this week)

2. Review (pay special attention to the corrections to the previous problems), and get prepared for the midterm exam on 10/27. Good luck!

Course Roadmap:

In the first half of the fall semester, we have gone through the “counting part” (Can you remember the “counting strategies” we have learned so far? For example, the complementary method vs. the constructive method? Correcting the overcounting strategy, and so on). After the midterm exam, we will discuss “probabilities”. Have fun!

Chapter 5: More with Combinations

Lesson Review

Paths on a grid: The solution to the general problem is if you must take X right steps, and Y up steps then the number of routes is simply the ways of choosing where to take the right (or up) steps. i.e.

Combine the counting techniques: complementary counting or casework; overcounting strategies, and so on.

Distinguishability matters (see section 5.4, examples of the balls and boxes).

Homework

Page 83, Ex 5.2.1 (classwork), 5.2.2, 5.2.3,

Page 88, Ex 5.3.1, 5.3.2, 5.3.3, 5.3.5

Page 91, Ex 5.4.1 (classwork), 5.4.2, 5.4.3

Page 93, Ex 5.10

Extra Resources

I am very glad to see that our students are making steady progress, especially in the quality of the homework. Keep up the good work!

Look forward to meeting you all next week. We miss our dear friends Annabelle, Jasmine, and Julia today:)

A warm welcome to our new friend, Susan!

Happy Birthday to Linda (10/5)!

Lesson review:

Chapter 4: Committees and Combinations

1. Choosing a “committee” (“combination”): the order doesn’t matter; while in a “permutation”, the order matters.

2. “n choose r”: 

3. Combinatorial identity:

Homework:

• Page 68: Ex 4.2.1, 4.2.3, 4.2.4
• Page 72: Ex 4.3.2 to 4.3.5
• Page 75: Ex 4.4.1 to 4.4.4

Extra Resource:

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb#x9e81a4f98389efdf:combinations

Next week:

if you are not coming next Sunday, please inform me ahead. So that I can plan my lesson accordingly. Thanks 🙂

AOPS-CP Week 4: Chapter 3: Correcting for Overcounting

Lesson Review

1. Permutations with repeated elements:

https://www.ck12.org/probability/permutations-with-repetition/lesson/Permutations-with-Repetition-BSC-PST/

https://www.mathwarehouse.com/probability/permutations-repeated-items.php

2. Counting pairs of items (understand why we need to divide by 2)

Formula to find the sum of n numbers (understand how we derived it step by step).

1+2+3+…+(n-1) + n = n (n+1) /2

3. Counting with symmetries

a) rotational symmetry (round table seats) = n!/n = (n-1)!

b) reflective symmetry (flipping over, key chain) = n!/2n = (n-1)!/2

Homework:

Page 52: 3.2.1, 3.2.2., 3.2.3, 3.2.5

Page 57: 3.3.1, 3.3.3, *3.3.5 (it’s a bonus, but not difficult. Just try!)

Page 61: 3.4.1, 3.4.3

Final Tips:

Please do not try to remember the formulas by heart, instead, try to understand them, and learn how to derive them.

We value procedures. Please show all your steps/procedures in math problem solving: both in classwork and homework. You will benefit from it in the long run.

Lesson Review:

We introduced 4 basic counting techniques today:

1) Casework: to divide the problem into different subcases, count each, and add the counts altogether. It helps break a counting problem into manageable pieces, but requires you to be organized and careful (do not omit any outcomes).

2) Complementary Method: what we want = total = what we do not want. n(A) = n(U)-n(A’)

3) Constructive Method: count how many ways we can construct a particular item.

4) Counting with restrictions: it is generally best to deal with the most severe restrictions first.

Some tips:

1) independent choices: multiply the number of options at each step; exclusive options: add the number of options.

2) Sometimes when you get stuck at complicated problems, try to start with some simple cases, and find a pattern, or hint, to help you solve the “big problem”.

Homework:

Page 34: Ex2.2.1, 2.2.3, 2.2.5* (*bonus, highly recommended)
Page 38: Ex 2.3.1, 2.3.3
Page 41: Ex 2.4.1, 2.4.3
Page 44: Ex 2.5.1, 2.5.4

Thanks to Tiffany (William’s mom) for sharing the first two chapters of the E-book. Here are the screenshots of the problems. In case the textbook has not arrived by Sunday, students can use these to finish the homework.

Feel free to reach out if you have any question! See you on Sunday!

Lesson Review:

1. The number of items in the list: a, a+1, …, b, b-1, where a≤b are integers, is b-a+1 (understand how it is derived).

2. Understand Venn Diagram

3. Count multiple events: (the number of ways A can occur) times (the number of ways B can occur)

4. Permutation: selelct r items in order from a group of n items in P(n,r) ways:

P(n,r) = n x (n-1) x (n-2) x … (n-r+2) x (n-r+1) = n!/(n-r)!

5. Factoral: n! = n x (n-1) x (n-2) x … 3 x 2 x 1

Homework:

Page 5: Ex 1.2.1, 1.2.2, 1.2.6

Page 12: Ex 1.3.1, 1.3.3

Page 18: Ex 1.4.1, 1.4.3, 1.4.5

Page 21: Ex. 1.5.1 (a) (c), 1.5.2