Based on the information given, you can draw the following Venn Diagram:

Now, you must begin by solving for . You know that the two circles together will have  in them. This is arrived at by subtracting the people who have neither books nor pens () from the "universe" of people in the sample space (). Now, we know that . This is because of the overlap of  in both groups. We have to get rid of one instance of that. Thus we can solve for :

Now, we can find the number of people with only books by subtracting  from the  to get .

Carter would not fall in set A, since he was not a President born in Virginia. 

He would not fall in B, since he was born before 1850.

He would fall in C, since his first name is James.

He would fall in the region included in set C, but not A or B - this is Region V.

Carter would not fall in set A, since he was not a President born in Virginia. 

He would fall in B, since he was born after 1850.

He would fall in C, since his first name is James.

He would fall in the region included in sets B and C, but not A - this is Region III.

The subset must comprise words that fall inside set , but neither nor .

Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.

The only set among the five choices that matches this description is the set

{price, value, pinna, trove, three}.

The subset must comprise words that fall inside sets  and , but not . Therefore, all of the words in the subset must begin with a consonant, end with a vowel, and not have six letters.

Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

: 

: 

: 

Taking these together, you get:

Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal   or . This is the total amount in the "universe" () minus the amount that is found outside of the two circles ().

Because the overlap happens once in each circle, you know that:

Given your data, you know:

Solving, this means:

Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal  or . This is the total amount in the "universe" () minus the amount that is found outside of the two circles ().

Because the overlap happens once in each circle, you know that:

Given your data, you know:

Solving, this means: . The overlap is the whole of circle !

Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal   or .  This is the total amount in the "universe" () minus the amount that is found outside of the two circles ().

Because the overlap happens once in each circle, you know that:

Given your data, you know:

Solving, this means:

Do not be tricked by this question! In order to solve for the overlap, you need to know the amount that is in the area outside of the circles but still inside the universal box area! You cannot figure out the answer without knowing this fact; therefore, you must select "No answer is possible."  We know that the two circles do not exhaust the universe because . This is not large enough to fill the complete . If it were, you would know that the overlap is .