#PLease do not post homework
Kenneth Chan
Math 454
HW 2

1.Let a,b,c be the number of walks from [0,0,0] in a 3D "MAnhattan Lattice". We want to prove that the explicit formula is true. (a+b+c)!/(a!*b!*c!).
To understand the numerator we look at a simple permutation. Let us say we want to organize abc as many ways possible, we can organzie it abc,acb,bac,bca,cba, and cab.
There are 6 ways also know as 3!. (a+b+c)! gives us the total numbers of permutations taking all paths with no orders. But that is impossible because we cannot 
walk up to floor 3 without walking up to floor 2 first. So we have the denominator to factor those paths out. We have a! number of ways we can take the different paths but only taking the path from start to finish makes
logical sense so we divide it by a!. Thus we divide the rest by b! and c! so we know we are moving in a straight line and not jumping around the paths. 
2. Even in k-dimensionals for the Manhattan lattice, the combination formula removes any impossibles paths letting us know the only logical paths due to the dividing of any paths that cannot work. The 
only difference with the alphabet is that all words with the same i's in the word are being removed. having a word [aiii] is the same as having a word [iaii] becasuse they both have three i's. For this problem we would have 
the total number of possible words the can be created 26^k divide by i factorial.
3.