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 .

In the range of 1 to 10, the prime numbers - the numbers with exactly two factors, one and itself - are 2, 3, 5, and 7. Therefore, there are six nonprime numbers (including 1, which, having only one factor, 1 itself, is considered to not be a prime number) represented by two balls each, and four prime numbers represented by three balls each. Therefore, the box will contain 

 balls.

Since 2 of the balls have a "1" on them - and 22 do not - the odds against drawing a ball with a "1" are 

, or 11 to 1.

In the range of 1 to 10, the prime numbers - the numbers with exactly two factors, one and itself - are 2, 3, 5, and 7. Therefore, there are six nonprime numbers (including 1, which, having only one factor, 1 itself, is considered to not be a prime number) represented by  balls each, and four prime numbers represented by  balls each. Therefore, the box will contain 

balls. Since 2 is prime,  balls will have a "2" on them, making the probability of drawing one 

.

Each of the ten integers is represented by three balls; each of the five vowels, two balls; each of the twenty-one consonants, one ball. Therefore, the total number of balls in the box is

.

Of these balls, three have a "1" on them; since "A" is a vowel, two have an "A" on them. Therefore, the probability of drawing a ball with either is .

Roshonda likes Shania Twain, putting her in set .

She dislikes one or both of Avril Lavigne and Kelly Clarkson. The set of people who like Avril Lavigne comprise ; the set of people who dislike Avril Lavigne comprise the complement of this set, . Similarly, the set of people who dislike Kelly Clarkson is . She falls in either or both of these sets, putting her in their union, .

Since she is in both  and , she is in their intersection, 

.

The correct choice is 

For a person to fall in the set  - that is, the intesection of  and  - (s)he must fall in both  and .

To fall in , the person is in either , , or both; in oher words, the person must like either The Allman Brothers Band or The Beach Boys, or both. 

To fall in , the person must fall outside ; in other words, the person must not like Crosby, Stills, and Nash.

John and Mary, both of whom like The Allman Brothers Band, fall in . However, Mary likes Crosby, Stills, and Nash, so she does not fall in . John does not like Crosby, Stills, and Nash, so he does fall in . Only John falls in the set .

First, there are two ways to seat the Smiths; Mr. Smith at seat 1 and his wife at seat 5, and vice versa.

Second, the number of ways to seat the remaining six guests is the number of ways to permute 6 from 6:

Multiply: