First, it is necessary to choose the couples that are each table. There are

 ways to choose the two couples that will be at the left table; this is equal to

.

Once those two couples are chosen, the remaining two couples will be at right.

For each table, there are 

ways to seat the four chosen persons.

By the multiplication principle. there will be 

 different arrangements that seat all four couples together.

We can apply the multiplication principle here. First, there are two possible places Rachel can occupy. One this is decided, there are two ways to seat her parents - with Mr. Jones to her left and Mrs. Jones to her right, and vice versa. The number of ways to seat the remaining five guests in the remaining five seats is 

.

Therefore, the number of arrangements fulfilling the requirement is

.

For each candidate, add the two day totals:

Allen: 

Paul: 

Veronica: 

Wendy: 

Allen was the highest vote-getter. However, Allen got 157 votes, and his opponents got a total of  votes, so Allen did not get a majority. He will fac the second-highest vote-getter, Veronica, in a runoff.

Ahmad received 193 votes plus 21.7% of the previously uncounted 115 votes, for a total of

 votes.

Michiko received 598 votes plus 42.6% of the previously uncounted 115 votes, for a total of

 votes.

Quinn received 210 votes plus 18.3% of the previously uncounted 115 votes, for a total of

 votes.

Zane received 189 votes plus 17.4% of the previously uncounted 115 votes, for a total of

 votes.

Michiko got the most votes with 647. Her opponents won a total of

 votes, more than Michiko, so Michiko did not win a majority. She will face the second-highest vote-getter, Quinn, in a runoff.

A person in the set - the union of the sets  and  - would have to fit either or both of the following descriptions:

The person could be in the set , in which case (s)he speaks Arabic. Jim speaks Arabic, so he is in ; consequently, he is in . Julie does not qualify for inclusion in this set this way.

The person could be in set , which means (s)he is in both , the complement of , and , the complement of . (S)he would have to speak neither Dutch nor Spanish. Julie speaks Dutch, so she does not qualify for inclusion in this way either.

 is the set of dancers. , the complement of , is the set of persons not in  - that is, the set of persons who are not singers.  is the intersection of the two, or the set of all persons in both  and ; to be in , a person must be a dancer and must not be a singer. Mary is excluded from  because she is a singer; Florence is excluded because she is not a dancer.

By DeMorgan's Laws, 

That is, the complement of the union of the sets is the intersection of the complements of the sets. Therefore, for a person to be in this set, it is necessary for the person to be in the complements of both  and  - that is, the person cannot be in either set. Any person who can play spades or hearts must be excluded, and since both Beyonce and Aaliyah can play at least one game, both are excluded.

 is the intersection of two sets. One is the complement of , which is the set of all elements not in ; these are the numbers that are not prime. The other is , the set of all odd numbers. Therefore, the correct choice must be odd and not prime. 

2 and 32 are eliminated for being even. Of the two odd choices, 17 is a prime number, having only 1 and 17 as factors. 39 has more than two factors (3 is a factor, since 39 divided by 3 yields quotient 13), so it is not prime. 39 is the correct choice.

There are five vowels, each represented by three balls; the number of balls with vowels is 

.

There are twenty-one consonants, each represented by two balls; the number of balls with consonants is

.

The odds against drawing a vowel are therefore 

, or 14 to 5.

There are five vowels, each represented by  balls, so there will be  balls with vowels; each of the twenty-one consonants will be represented by one ball, so the total number of balls is . Since there are  balls with an "E" on them, the probability of drawing one of these balls is .