The grayed portion of the Venn diagram corresponds to those integers which are not in any of $\displaystyle A$, $\displaystyle B$, or $\displaystyle C$. Therefore, we eliminate any choices that are in any of the three sets.

$\displaystyle B$ is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.

$\displaystyle C$ is the set of perfect square integers; we can eliminate 144, since $\displaystyle 12^{2} = 144$.

$\displaystyle A$ is the set of integers which, when divided by 4, yields remainder 2. Since $\displaystyle 154 \div 4 = 38 \textrm{ R }2$ , we can eliminate 154.

All four choices have been eliminated.

$\displaystyle A \cup B$ is the union of sets $\displaystyle A$ and $\displaystyle B$ - that is, the set of all elements of $\displaystyle U$ that are elements of either $\displaystyle A$ or $\displaystyle B$. We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except $\displaystyle c$. Therefore, 

$\displaystyle A \cup B = \left \{ a,b,d,e,f,g,h\right \}$

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

In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: $\displaystyle 155-55 = 105$. Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that $\displaystyle x$; therefore, we know that:

$\displaystyle 78+55-x = 105$

Solving for $\displaystyle x$, you get:

$\displaystyle 133 - x = 105$

$\displaystyle - x = -28$

$\displaystyle x = 28$

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

To solve this, remember that the total number of values in the two circles is:

$\displaystyle 125+57 - 22 = 160$

(We must do this because of the overlap.  You need to subtract out one instance of that overlap.)

If we assign the value $\displaystyle x$ for the unknown region, we know:

$\displaystyle x + 160 = 310$

$\displaystyle x = 150$

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

It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely $\displaystyle 45 - 20 = 25$. We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).

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

Now, you must begin by solving for $\displaystyle x$. You know that the two circles together will have $\displaystyle 105 - 20 = 85$ in them. This is arrived at by subtracting the people who have neither books nor pens ($\displaystyle 20$) from the "universe" of people in the sample space ($\displaystyle 105$). Now, we know that $\displaystyle Books + Pens - x = 85$. This is because of the overlap of $\displaystyle x$ in both groups. We have to get rid of one instance of that. Thus we can solve for $\displaystyle x$:

$\displaystyle 45 + 57 - x = 85$ 

$\displaystyle 102 - x = 85$

$\displaystyle -x = -17$

$\displaystyle x = 17$

Now, we can find the number of people with only books by subtracting $\displaystyle 17$ from the $\displaystyle 45$ to get $\displaystyle 28$.

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 $\displaystyle C$, but neither$\displaystyle A$ nor $\displaystyle B$.

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 $\displaystyle A$ and $\displaystyle B$, but not $\displaystyle C$. 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}.