Section+4.7+Part+2

Today in class we discussed several problems math 2^n math
 * (including the train problem)
 * isomorphism of problems (in particular, permutation and combination problems)
 * a new choose notation definition of

The all-too familiar train problem begins with the question of how many distinct trains of length //n// can one make, using a row of rods (with integer lengths) that combine to have length //n//.

The simplest example that we began with in class was a train of length five, which had combinations 1-1-1-1-1, 2-1-1-1, 1-2-1-1, 1-1-2-1, 1-1-1-2, 1-2-2, 2-1-2, 2-2-1, 3-1-1, 1-3-1, 1-1-3, 3-2, 2-3, 4-1, 1-4, and 5. That comes out a total of 16 trains of length 5. As Ms. Tyson taught us in class, the total number of trains can be rewritten as



or more generally,



We also went on to discuss a choose notation definition of, i.e.,

Most importantly, I think, we discussed whether or not problems are isomorphic.


 * Problem 1a.** "You have nine books, but there is only room for four of them on your shelf. In how many different ways can you line up four of the nine books on your shelf" (CME 2008)?

Problems where you have //x// number of things and n (x>n) places to put them can be nicely envisioned by placing the number of available slots you have, which is four in our case.

_ _ _ _

We know that at first there are 9 books, one of which can be placed in one of four slots--but this can be done for nine different books.

9 _ _ _

For the second slot, we know that there are 8 remaining books, one of which can be placed in slot 2, and 3 slots left. There are 8 ways the remaining books can be placed in slot 2. Continuing this pattern, there are 7 ways the remaining books can be placed in slot 3, and 6 ways the remaining books can be placed in slot 4. Because we can envision this process being drawn as a tree diagram, we know that the numbers must be multiplied, thus we get:

ways. This question is more complicated than a simple combination problem because instead of asking you how many ways you can choose some amount of books from the total amount of books, it asks you how many ways you can choose some amount of books from the total amount of books and place them into some particular arrangement.


 * Problem 1b.** "In how many different ways can a person choose four books from a pile of nine distinct best-sellers" (CME 2008)?

This question is literally solved by doing ways, because this says how many ways (126) you can pick some amount of books (4) from a larger amount (9). Subtly different from **Problem 1a**.

--These problem's are **not** isomorphic


 * Problem 2a**. "There are 14 airports in a country. You can go from any one of them to any other airport in the country by a direct flight. How many direct flights are there" (CME 2008)?


 * Problem 2b.** "How many diagonals does a 14-sided polygon have" (CME 2008)?

While these problems appear isomorphic, they too are not. Look at the picture of the 14-gon to the left, and imagine that the airports are set up at each of the 14 vertices of the polygon. Quite simply, the 14-gon does not allow you to draw a diagonal between the points adjacent to it (allowing you to make 11 diagonals, subtracting the starting point), whereas the airport problem lets you fly to adjacent airports. There are 14! direct flights as opposed to the (total number of vertices minus adjacent and starting vertices) * total remaining vertices=total number of diagonals in the diagonal problem, where there are 11 vertices to travel to * per all 14 vertices=154 diagonals minus double-counted diagonals. One answer is more than 1 trillion while the other is under 200. These problems are **not** isomorphic.

The last problems we shall look at are actually isomorphic:


 * Problem 3a**. "In how many ways can you write 20 as a sum of three counting numbers" (CME 2008)?


 * Problem 3b.** "In how many ways can you put 20 quarters into three colored pockets (red, green, and blue) so that there is at least one coin in each pocket" (CME 2008)?

These problems are isomorphic because they both ask how many different ways things can add up to equal 20. In the case of **Problem 3a.**, it is the three numbers that equal twenty, and in the case of **Problem 3b.**, it is the number of quarters in each pocket that add up to twenty; both questions answer the same question essentially.

These questions **are** isomorphic.

Works Cited:

University of South Florida. "Polygon consisting of 14 sides." 2008. http://etc.usf.edu/clipart/37300/37390/14-gon_37390_lg.gif GIF Image.

CME Project Development Team. //Precalculus//. Upper Saddle River: Pearson, 2008. Book.