In the range of 1 to 10, the composite numbers - the numbers with at least one factor other than 1 and itself - are 4, 6, 8, 9, and 10. Therefore, there are five composite numbers represented by two balls each, and five noncomposite numbers (1, 2, 3, 5, 7) represnted by 1 ball each. The total number of balls is

Among the odd numbers, 1, 3, 5, and 7 are represented by one ball each; 9 is represented by two, for a total of 6 balls out of 15. This makes the probability of drawing an odd number

The following choices need to be made:

First, the decision must be made as to the table at which the Johnsons will sit. There are two choices.

Next, the decision must be made as to which seats the Johnsons will occupy at that table. There are two choices - the "north-and-south" seats and the "east-and-west" seats.

Third, the decision must be made which one will sit where; there are two choices now.

Fourth, the decision must be made as to how the six children will be seated - this will be

.

By the multiplication principle, the number of seating arrangements with the Johnsons sitting across from each other will be

An integer is divisible by 11 if and only if its alternating digit sum is divisible by 11. If we let  be the missing digit, then the alternating digit sum is 

Through trial and error, we see that no value of  from 0 to 10 makes this a multple of 11. This makes none the correct choice.

Let , , , and  represent the set of students who like "The Walking Dead", "Boardwalk Empire", "The Sopranos", and "Deadwood", repsectively.

Everyone who likes "The Walking Dead" likes "Boardwalk Empire", so .

Everyone who dislikes "The Sopranos" likes "Deadwood", so .

Nobody likes both "Boardwalk Empire" and "The Sopranos", so  and  have no elements in common - they are disoint sets. One way of saying this is that everyone who likes "Boardwalk Empire" dislikes "The Sopranos" - that is, . 

These statements can be put together as follows:

We can represent Philip as . Philip likes "The Walking Dead", so it follows that

, which means he does not like "The Sopranos",

and

, which means he likes "Deadwood".

Let , , , and  represent students who like Diddy, Prince, Weezer, and Bruce Springsteen, respectively.

Every student who likes Diddy also likes Prince, so .

Every student who likes Weezer also likes Bruce Springsteen, so 

No student likes both Prince and Bruce Springsteen, so  and  have no elements in common - they are disjoint sets.

A Venn diagram for this scenario is below:

Note that since Jerry ("j") is an element of , he cannot be an element of  or . Therefore, he does not like Diddy or Prince.

The number of miles per hour over the limit is the difference of the number of miles per hour the driver was going and the speed limit - this is . The fine is $75 plus $4 times this difference:

 is the intersection of two sets,  and .  itself is the intersection of , the set of persons who like The Rolling Stones, and , the set of persons who like Alanis Morissette.  is the set of persons who like Bob Seger, so , the complement of , is the set of persons not in  - that is, persons who dislike Bob Seger.

Therefore, anyone in the set  likes The Rolling Stones, likes Alanis Morissette, and dislikes Bob Seger. The correct choice is "John likes The Rolling Stones and Alanis Morissette, but he dislikes Bob Seger."

First, we note that the total number of seating arrangements for eight people is 

The number of arrangements that would have Steve and Jim in adjacent seats would be calculated as follows:

The number of ways to choose two adjacent seats is eight: 1-2, 2-3, and so forth up to 8-1. Multiply this by 2, since each choice has two different orders for Steve and Jim. Multiply this by 

,

the number of ways the remaining six persons can be seated. This makes

seating arrangements with Steve and Jim together. Since this is not desired, subtract this from the total number of arrangements:

This is the number of arrangments without Steve and Jim together, and it is the correct choice.

By DeMorgan's Law,

That is, the complement of the union of two sets is the intersection of their complements. 

Also, the complement of a complement of a set is the set itself, so  and . Therefore, 

.

Consequently, we are looking for a number that falls in the intersection of  and . The number must be both prime and odd. 2 and 4 are eliminated for being even. 1 is not considered a prime number, having only one factor. 3, which has only two factors, 1 and 3, is prime, and is the correct choice.

A number that is divisible by 44 must also be divisible by all factors of 44.

One factor is 4. The number is divisible by 4 if the last two digits form a number itself divisible by 4. Since the last two digits are "12", which is divisible by 4, we know the number to be divisible by 4 regardless of the digit chosen.

Another factor is 11. The number is divisible by 11 if and only of the alternating sum has an absolute value divisible by 11. If  is the missing digit, then 

.

We can set each of the choices equal to and see which one works:

Only in the case is the alternating sum divisible by 11, so this is the correct choice.