The symbol  stands for the union between two sets.  Therefore,  means the set of all numbers that are in either A or B.  Looking at our choices, the only number that isn't in either A, B, or both is 23.

We can draw a Venn diagram to see these two sets of students.

We need to find the overlap between these two sets. To find that, add up the total number of students who are taking history and the total number of students who are taking calculus.

Notice that we have more students this way than the total number who were polled. That is because the students who are taking history AND calculus have been double counted. Subtract the total number of students polled to find out how many students were counted twice.

The notation  stands for "union," which refers to everything that is in either set.  refers to the group of students taking either Calculus or Spanish (everyone on this diagram except those taking only Biology). From the diagram, Patrick and Ashley are the only students taking neither Calculus nor Spanish, so Patrick is the correct answer.

We can draw a Venn diagram of these students.

Drawn this way, there are more students on the Venn diagram than we have.

This is because some of the students play both sports and should be in the overlap on the Venn diagram. To find the number of students in the overlap, subtract the total number of students given from the number on the diagram.

This represents the number of students who were counted twice, or the number in the overlap.

We can redraw the correct Venn diagram with this number.

The notation  stands for "A union C," which refers to everything that is in either set  or set .

When we add the numbers together, we get:

A Venn diagram can help us determine the total number of students in the class.

First, we must calculate the number of students who have ONLY cats or ONLY dogs. First, for cats, 15 students have cats, and 5 students have both cats and dogs.

Ten students have only cats.

For dogs, 12 students have dogs, and 5 students have both cats and dogs.

Seven students have only dogs. 

Using this information, we can fill in the Venn diagram.

This diagram shows the 10 students with only cats, the 7 students with only dogs, the 5 students with both, and the 8 students with neither.  Adding up the numbers will give us the total number of students.

There are a total of 7 + 5 + 3 + 6 = 21 marbles. The probability of picking a yellow marble is 7/21 = 1/3. Then we put it back and choose a blue marble with probability 3/21 = 1/7. We do NOT put this blue marble back, but then we grab for a white. The probability of picking a white is now 6/20 = 3/10, because now we are choosing from 20 marbles instead of 21. So putting it together, the probability of choosing a yellow marble, replacing it and then choosing a blue and a white, is 1/3 * 1/7 * 3/10 = 1/70.

Add up the total number of jellybeans, 20 + 45 + 30 + 5 = 100. 

Divide the number of watermelon jellybeans by the total: 20/100 and reduce the fraction to 1/5.

There are four different types of cake. In this type of problem we want to guarantee we have one of each, so we need to assumbe we have very bad luck. We start with the red velvet since that is the type with the most cakes. If we open those 6 we are not guaranteed to have different ones. Then say we opened all five chocolate cakes, then all four carrot cakes. We still have only three types of cakes but opened 15 boxes. When we open the next box (16) we will be guaranteed to have one of each.

The probability of the point being inside the circle is the ratio of the area of the circle to the area of the square. If we suppose that the circle has radius r, then the square must have side 2r. The area of the circle is πr² and the area of the square is 〖(2r)〗²=〖4r〗², so the proportion of the areas is (πr²)/〖4r〗²  =π/4.