4.9+Intro+to+Binomial+(1-3)

Multiplying with distinct factors:
media type="custom" key="11994093" media type="custom" key="11994103" media type="custom" key="11994701"

At this point we are starting to get bored and annoyed with the annoying math... so let's start thinking about how to automatize this so I don't have to do as much work...

media type="custom" key="12019765"

So what we notice is that
 * The number of terms in the answer is doubling as we increase the number of factors on the original side.
 * In English speak: each term from the answer in the previous problem is multiplied by an media type="custom" key="11994485" and a media type="custom" key="11994487" to create two new terms in the answer to the new problem.
 * In Math speak: media type="custom" key="11994545"
 * Remember the media type="custom" key="11994521" means the product in the same way that media type="custom" key="11994525" means the sum.
 * In fact the number of terms in the answer is media type="custom" key="11994621"where media type="custom" key="11994627"is the number of factors in the question. (Later we'll look at a proof for this)
 * We also looked at this a little bit with the idea of lattice multiplication and the number of boxes that would need to be filled in.


 * The number of factors in each term in the answer is the same as the number of factors on the original side.
 * This also follows from the previous discussion as we see that in the first example media type="custom" key="11994555" each term in the answer has one factor and each time we move to another factor in the original question, we add another factor to each of the terms.
 * In fact if we look carefully at our answers we will see that each term in the answer has one factor from each of the factors in the original (or put another way each term has each possible subscript exactly once).


 * I had to learn a lot of new TeX commands

WHAT HAPPENS IF WE ERASE THE SUBSCRIPTS.... (This is really what the binomial theorem is about)
Let's look just at media type="custom" key="11994675"


 * 1) I can just do the problem out: media type="custom" key="11994695"
 * 2) I can look at my answer from the subscripts above and then erase the subscripts:media type="custom" key="11994711" becomes media type="custom" key="11994735"

So now what patterns do I see

 * The number of terms in the answer is one more than the exponent on the original question.
 * The degree of each term in media type="custom" key="11994765" is n (the exponent) because we saw in the subscripted case that the number of factors in each term in the answer was the same as the number of fcators in the question
 * Each type of degree n term must occur because of the total number of terms in the subscripted case.
 * Therefore we have every possible exponent of media type="custom" key="11994777" (0, 1, 2, 3, ... n) appearing which gives us media type="custom" key="11994783" terms in the answer.


 * The coefficients are the same as Pascal's triangle.
 * The coefficient on media type="custom" key="11994895" is media type="custom" key="11994913"because I am choosing which one of the original three factors is going to provide me with a b.
 * When I had subscripts I had media type="custom" key="11994919" which shows how I need to choose which factor is providing the b.


 * The sum of the coefficients is media type="custom" key="11994947"
 * I had that many terms when there were subscripts and all those terms are still there some were just combined which created the coefficients
 * That the sum of a row of Pascal's triangle from earlier so that's the sum of the coefficients.

SO now what?

 * Read media type="custom" key="11994973"4.9 to see how this connects to the number of ways that Mrs. Pascal can get to work
 * Think about what it would be helpful to do to improve the wiki.
 * New naming convention: Section # Helpful Title (Mo - Day)