Because order is not important in this problem (i.e. chocolate chip, pecan, butterscotch is no different than pecan, butterscotch, chocolate chip), it is dealing with combinations rather than permutations.

The formula for a combination is given as:

where is the number of options and is the size of the combination.

For the ice cream choices, there are thirty-one options to build a three-scoop sundae. So, the number of ice cream combinations is given as:

Now, for the topping combinations, we are told there are ten options and that Mohammed is allowed to pick two items; however, we are also told that Mohammed has already chosen one, so this leaves nine options with one item being selected:

So there are 9 "combinations" (using the word a bit loosely) available for the toppings. This is perhaps intuitive, but it's worth doing the math.

Now, to find the total sundae combinations—ice cream and toppings both—we multiply these two totals:

Step 1: Write in multiplication form:

Step 2: Multiply Out

The subset symbol should be read as "is a subset of" So it's the first letter is a subset of the second letter. To be a subset, all of its elements must be contained in the other set.

The only one of these relationships that is true (where the entire set on the left is in the right set) is

The elements of B, 4, 5, 6, and 8 all appear in set D.

Step 1: Define a subset. A subset of a bigger set is a smaller part where all the elements in the subset must be present in the bigger set.

The answer is . All numbers in this subset are in the bigger set, which is provided in the question...

Step 1: Recall the definition of a subset. A subset is a small part of a bigger given set; ALL numbers in the subset must be in the original set.

Step 2: We look at the original set and then look at the answers. If there is any answer that has an element that is not showing up in the original set, than that is the right answer.

Let's look at: . The is in the original set, but the is not. Since we have a number in the subset that is not in the original set, we can say that is not a subset.

The other two answers with brackets are subsets of the original set. All elements in the subsets are present in the original set.

The empty set is always a subset of any given set.

The answer is

Step 1: Define a subset

A subset of a big set is a set that has some of the elements that are in the bigger parent set.

The answer is because both and are both in the bigger set.

Step 1: A subset of a set must have elements in Set A. If any number in the subset is not in the original set, then that subset is not a subset of that set.

{} is not a subset because is not in Set A.

For this problem, since the order of the vases matters (red blue yellow is different than blue red yellow), we're dealing with permutations.

With selections made from potential options, the total number of possible permutations(order matters) is:

Step 1: Arrange the numbers in the set from smallest to largest:

After we arrange, we get: .

Step 2: To find the range, subtract the largest number from the smallest number.

The range of the data set is .

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

This is written as

There are possible groups of 3.