Factorial numbers such as are evaluated as follows;

So to evaluate the expression

(Start)

(Expand the factorials)

(Cancel the common factors)

(Simplify)

This combination problem asks us the number of ways to arrange four letters, including two that are the same. Four different letters can be arranged in  different ways, but since we have two of the same letter, , we have to divide  by . In any combination problem, if we have a total of  letters, then for every  number of the same letters, we have  .

180 is the correct answer. This problem asks for the number of possible permutations of these six letters, which include two letters that are repeated twice,  and . In any combination problem, if we have a total of  letters, then for every  number of the same letters, we have  . So, this problem's situation can be modeled by the expression . We take the factorial of  because we have six letters to be arranged. We divide by  because we have two instances of letters being repeated twice.

Since the four letters are different, they can be arranged in  different ways. 

This problem is asking us in how many ways the four different presents can be "arranged," or in this case, given to different people. Therefore, the answer is :

The first thing that we should do is break down the different group of same letters in Tennessee: We have one "T," four "E's," two "N's," and two "S's." We have a total of nine letters; four of these are the same, as are two more pairs. In any combination problem, if we have a total of  letters, then for every  number of the same letters, we have  . So, we can model this problem's situation with the expression :

This question asks us to count the number of ways to arrange 3 of the same items and 2 different items. Let's assign letters to each of them: "" for one of the bicycles,  for the first type of skateboard, and  for the second type of skateboard. In this notation, our set looks like this: 

We have a total of five individuals to receive five items, one item each. In any combination problem, if we have a total of  letters, then for every  number of the same letters, we have  . So, for this situation, the total is given by dividing the total number of ways to arrange the items 5! by 3!, since 3 of these items are the same:

Here, the number of "slots" is different than the number of letters we can use. We must first determine in how many ways we can select the letters to be arranged in our three-letter combination. Because we are selecting some but not all out of a presented group of options and order is not important, we can use the formula , where  is the total number of things from which we are choosing and  is the number things we are selecting for each group. In this case,  and , so our equation will look like this:

We have 36 characters to choose from, since there are 10 single-digit numbers (0-9) and 26 letters in the alphabet. The length of the code is two characters and nothing was said about repeated combinations not being allowed, so the correct answer is given by , which is .

In this problem, we are selecting four letters from a group of five letters that contains a pair of duplicate letters: two s. We can model this situation using the formula , where  is the number of things we have to choose from and each value of  corresponds to the number of duplicate items in a given group. In this case, we have five letters to pick from, so , and two of them are duplicates, so . This makes our equation: