Section+4.2

1) How many three different numbers can be made out out of 1s and 2s? 2) How many shape/color combinations can be made out of three shapes and two colors? 3) How many different pizza's can be made using three different toppings? 4) How many possible outcomes are there if a coin is flipped three times?
 * In class we looked at four different problems:**


 * Ways in which we initially solved the problems for example one:**

__Write out all the combinations__: 111 112 121 211 122 212 221 222

__Make a tree diagram__:

__Think about it__: There are 3 slots to fill (3 place values) and 2 choices for each (1 or 2). Therefore the problem can be solved by taking 2 to the third power.


 * Ways in which we initially solved the problems for example two:**

__Write out all the combinations__: plain o m p om op mp omp

__Think about it__: There are 3 toppings (which can be thought of as 3 slots) and 2 choices for each topping (on or off)


 * We found that the answers to all the problems were the same. Why?**

It turns out that the problems were all mathematically equivalent or, isomorphic. Therefore, in fact, all of them can be expressed mathematically in the same way. In every case there were essentially 3 slots and 2 choices or states for each slot.

Going back to the question: How many three different numbers can be made out out of 1s and 2s? We can classify each state by the number of 2s in it. Therefore there are four groups: zero 2s, one 2, two 2s, two 2s, and three 2s. From here we can look at how many states are in each group and find the following pattern: 111 - there is one case where there are no zeros, which is equal to: (0 choose anything is 1 because there is one way to chose nothing)

112 121 - there are three cases where there is one 2, which is equal to: 211 (it is impossible to format this any better)

122 212 - there are three cases where there are two 2s, which is equal to: 221

222 - there is one case where there are 3 2s, which is equal to:

From here we can see the connection to Pascal's triangle:

(note that the triangle on left has six rows and the one on the right has five)

Image credits: wikipedia and mathsisfun.com