For Monday, March 9th, 2020

For Monday:

• In class we considered the following problem:  How many ways can pairs be formed from 2n total people?  We obtained an answer:  $(2n)!/ (n!2^n)$.  Give an argument that the result should also be ${2n}\choose{2}$${2n-2}\choose{2}$$\cdots$${4}\choose{2}$${2}\choose{2}$$/ n!$.  Are these really the same?  Can you give an algebraic proof that they are?
• Read Hammack, Sections 3.3 and 3.4 and do Exercises (this will take more than an hour, so continue when you have time).  Hammack goes into more detail that we did in class, but all the extra detail is really just more practice of the same ideas, and more ways to talk about the same ideas.  In some cases, he just gives names to the different types of things we encountered already:  permutations, k-permutations.  In other cases, you’ll see him discuss the same principles I talked about in class, but written out formally with set notation (Subtraction Principle, Addition Principle etc).  This is great practice in set notation!  Try to line up what he writes with what you learned in class.  So although he goes into more detail, I think it’s great practice to read these chapters actively.  But it will also take a while, so budget some time when you have it to finish this task.
• For pedagogical reasons, I plan to put Monday’s proof quiz off until Wednesday.  I’m hoping to do more examples of proofs that argue about counting.