Section+4.7+12-1

In Section 4.7, we talked about combinations.

The day before, we'd talked about permutations, which we defined as a //one-to-one function from a set to itself//**.** In a permutation, //order matters.// Conversely, in a combination, //order does not matter.// The official definition in the book is: //A subset of a set S is a combination of elements of S. You can think of a combination as being k things from from the set S, which itself has n elements.//
 * __Permutations vs. Combinations__**

__**In Class**__ Today, we spent class going over several problems relating to combinations in our groups and together with Mrs. Tyson.


 * Class Officers Problem Set**

There is a class of 20 students. I want to pick a president, vice president, and secretary/treasurer. How many ways can I pick three officers? Process: this is a straightforward permutation, because here order matters --
 * Question 1**



Again, there is a class comprised of 20 students from which I wish to pick three officers, but this time the three students I want to pick are //Interchangeable.// Process: because the class officers are interchangeable, here order does not matter.
 * Question 2**



The denominator is 17!3! because like the three L's in GULLIBLE, the three people are interchangeable. Another story to use to explain this problem is that you are choosing any three people from a group of 20, so 20 nCr 3 is another way to obtain your answer.

__**A General Rule:**__ If I have a group of n objects and I want to interchangeably pick k of them, the number of ways I can do that amount to the following --

Expanding on this, we get,

Using words to explain this algebraic proof, this makes sense because to make a subset of k objects from n objects, I can easily pick the elements that get left out of the set. (Say, for example, I want to pick those three students from a class of 20 -- If I pick 17 students from that class and exclude those three officers, I will logically get the same number of subsets).

Lastly, we proved using factorials that

because



For example: say there are five students (A, B, C, D, E) in a group, and you wish to choose three - there are 5 choose 3 ways to do this. Now, say you want to make a group of three while making sure to //exclude// student A - there are 4 choose 2 ways to do this. If you want to choose a group of three while making sure to //include// Student A, there are 4 choose 3 ways to do this. These two results sum to 5 choose 3.


 * __A General Rule:__**



Another way to think about this property besides the algebraic proof and the story above is to look at the symmetry of Pascal's triangle and note how the numbers to the top left and right of //n// add to //n.// Image Credit: Wikipedia