GVSU MTH 345 Portfolio Problems

Part A
One example of a partial order that is not a total order is comparing the hitting ability of different baseball players. Say that we compare these players by number of homeruns hit, RBI's, and batting average where player $A\preccurlyeq B$ if and only if $A$'s homeruns hit, RBI's, and the batting average is less than or equal to each of $B$'s. This ordering is reflexive because for any player $A$ there homeruns, RBI's, and batting average is equal to itself. This ordering is antisymmetric because it is impossible for two players $A$ and $B$ to have the same batting average and not be the same player.

Proof Wiki Definitions
Eulerian Graph

Characteristics of Eulerian Graph

Related Articles
Partial Order

A
First we place the L. We have 7 places to choose. Next we pick a vowel out of 5 and select a location out of 6 places. Next we pick a vowel out of 5 and pick a location for it out of 5 places. Pick one digit out of ten and a location for it out of the four remaining. Pick one digit out of ten and a location for it out of the three remaining. Pick one digit out of ten and a location for it out of the two remaining. Pick a letter that is not a vowel and put this letter in the last location.

By the multiplication rule we arrive at \[(1\cdot 7) \cdot (5 \cdot 6) \cdot (5 \cdot 5) \cdot (10 \cdot 4) \cdot (10 \cdot 3) \cdot (10 \cdot 2) \cdot (21 \cdot 1) = 2,646,000,000. \]

B
Pick a location out of seven for the first 7. Next pick a location out of six for the second 7. Next we pick from the group of letters and digits excluding the five vowels and the five even digits. We choose a spot for this out of the five remaining. We continue to do this the the last four spots.

By the multiplication rule we arrive at \[ (1 \cdot 7) \cdot (1 \cdot 6) \cdot (26 \cdot 5) \cdot (26 \cdot 4) \cdot (26 \cdot 3) \cdot (26 \cdot 2) \cdot (26 \cdot 1) = 59,882,135,040. \]