Discrete Mathematics
# Permutations

Joel has a set of 5 different horse figures and a set of 6 different train models. He has a shelf in his room that he is going to put them on, but he finds out he only has room to put either the entire set of horses or the entire set of trains on the shelf.

How many different ways can Joel arrange either all the horses or all the trains on his shelf?

Consider the two sets \( A = \{ 1, 2, 3, 4 \}\) and \(B = \{ 0, 1, 2, 3, 4 \}, \) and a function \( f: A \rightarrow B. \)

Find the number of functions \(f\) that satisfy \( f(1) + f(2) =2. \)

Find the number of ways to paint the faces of a regular tetrahedron using 4 different colors. All colors must be used.

**Details and assumptions**

Two colorings are identical if the tetrahedrons can be rotated to look identical.

It is happy hour on Friday. Sue, Sam, Pete, Calvin, David and Bradan are fooling around at their office desks. There are 6 desks, which correspond to where they sit during the day. How many ways are there for them to occupy a seat at the various desks, such that at most 1 of them is in the correct spot?

**Details and assumptions**

Each person sits at 1 desk. Each desk only allows for 1 person.

Lisa has 8 animal ornaments she wants to arrange on her shelf which has just 8 spots in a row. There are 2 mice, 2 dogs, 2 frogs, 1 giraffe, and 1 elephant. Lisa knows that elephants are afraid of mice, so she wants to arrange the ornaments so that one mouse is somewhere to the left of the elephant and the other mouse is somewhere to the right of the elephant, to trap the elephant between the mice. She also knows that frogs like to jump around, and hence wants the frogs to have an even number of spots between them. How many different ways can Lisa arrange the animals?

**Details and assumptions:**

The ornaments of the same type are indistinguishable.

0 is an even number. So the frogs having 0 spaces between them, i.e. the frogs being adjacent, is also allowed.

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