\(\displaystyle (R \cap M) \cap S'\) is the intersection of two sets, \(\displaystyle R \cap M\) and \(\displaystyle S'\). \(\displaystyle R \cap M\) itself is the intersection of \(\displaystyle R\), the set of persons who like The Rolling Stones, and \(\displaystyle M\), the set of persons who like Alanis Morissette. \(\displaystyle S\) is the set of persons who like Bob Seger, so \(\displaystyle S'\), the complement of \(\displaystyle S\), is the set of persons not in \(\displaystyle S\) - that is, persons who dislike Bob Seger.

Therefore, anyone in the set \(\displaystyle (R \cap M) \cap S'\) 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 

\(\displaystyle P (8,8) = 8! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 = 40,320\)

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 

\(\displaystyle P (6,6) = 6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720\),

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

\(\displaystyle 8 \times 2 \times 720 = 11,520\)

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

\(\displaystyle 40,320 - 11, 520 = 28,800\)

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

By DeMorgan's Law,

\(\displaystyle (P' \cup O') ' = (P')' \cap( O') '\)

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 \(\displaystyle (P')' = P\) and \(\displaystyle (O)' = O\). Therefore, 

\(\displaystyle (P' \cup O') ' =P \cap O\).

Consequently, we are looking for a number that falls in the intersection of \(\displaystyle P\) and \(\displaystyle O\). 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 \(\displaystyle A\) is the missing digit, then 

\(\displaystyle 7-2+A - 0+1-2 = 4+A\).

We can set each of the choices equal to \(\displaystyle A\) and see which one works:

\(\displaystyle A = 0: 4+A = 4+ 0 = 4\)

\(\displaystyle A =3: 4+A = 4+ 3 = 7\)

\(\displaystyle A =5: 4+5 = 4+5 =9\)

\(\displaystyle A =7: 4+A = 4+7 = 11\)

Only in the case \(\displaystyle A = 7\) is the alternating sum divisible by 11, so this is the correct choice.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 23^\circ F\) below zero and in Tucson it is \(\displaystyle 65^\circ F\) above zero; therefore we can say:

\(\displaystyle 23+65=88^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 23\) units away from zero and Tucson’s is \(\displaystyle 65\) units away.  

We can see that Tucson is \(\displaystyle 88^\circ F\) warmer than New Castle.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 27^\circ F\) below zero and in Tucson it is \(\displaystyle 70^\circ F\) above zero; therefore we can say:

\(\displaystyle 27+70=97^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 27\) units away from zero and Tucson’s is \(\displaystyle 70\) units away.  

We can see that Tucson is \(\displaystyle 97^\circ F\) warmer than New Castle.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 31^\circ F\) below zero and in Tucson it is \(\displaystyle 75^\circ F\) above zero; therefore we can say:

\(\displaystyle 31+75=106^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 31\) units away from zero and Tucson’s is \(\displaystyle 75\) units away.  

We can see that Tucson is \(\displaystyle 106^\circ F\) warmer than New Castle.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 33^\circ F\) below zero and in Tucson it is \(\displaystyle 80^\circ F\) above zero; therefore we can say:

\(\displaystyle 33+80=113^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 33\) units away from zero and Tucson’s is \(\displaystyle 80\) units away.  

We can see that Tucson is \(\displaystyle 113^\circ F\) warmer than New Castle.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 32^\circ F\) below zero and in Tucson it is \(\displaystyle 85^\circ F\) above zero; therefore we can say:

\(\displaystyle 32+85=117^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 32\) units away from zero and Tucson’s is \(\displaystyle 85\) units away.  

We can see that Tucson is \(\displaystyle 117^\circ F\) warmer than New Castle.

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is \(\displaystyle 25^\circ F\) below zero and in Tucson it is \(\displaystyle 90^\circ F\) above zero; therefore we can say:

\(\displaystyle 25+90=115^\circ F\)

Also, we can solve this problem by using a number line. New Castle’s temperature is \(\displaystyle 25\) units away from zero and Tucson’s is \(\displaystyle 90\) units away.  

We can see that Tucson is \(\displaystyle 115^\circ F\) warmer than New Castle.