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:

A whole number divisible by 24 is also divisible by any factor of 24, including 8.

A whole number is divisible by 8 if and only if its last three digits form a number which itself is divisible by 8. We can try each of the three digits: 

None of the choices yields a number divisible by 8. Since 8 cannot be a factor of this number,  neither can 24.

A number divisible by 36 must be divisible by any factor of 36; two such factors are 4 and 9, so we can test for divisibliity by both.

The number is divisible by 4 if and only if its last two digits form a number divisible by 4. 32, 52, 72, and 92 are all divisible by 4:

Therefore, all of the choices make the number divisible by 4. Therefore, we see which one makes the number divisible by 9, which happens if and only if the sum of the digits is a multiple of 9. If we let be the missing digit, then the digit sum is

.

We can test each digit by substituting for :

Since 18 is the only digit sum divisible by 9, 3 is the correct digit to fill in the box.

 is the complement of , making  the set of all persons not in  - that is, the set of all people who do not own a canoe.  is the set of all persons who own a motorboat. To be in , the union of the sets, a person must be in one or the other set, or both, or, equivalently, the person must either not own a canoe or must own a motorboat, or both. Donald is in  by virtue of owning a motorboat. Ed, however, owns a canoe, but does not own a motorboat, so he does not meet either qualification. 

First, the table at which Mr. Gregory is seated must be decided - there are two alternatives. Mrs. Gregory will be at the other table.

Then it must be decided which of the four seats Mr. Gregory will occupy at his table, and which of the four seats Mrs. Gregory will occupy at hers.

Finally, it must be decided how the six children will be arranged in the remaining six seats - this can happen in

ways.

Apply the multiplication principle; there will be 

possible arrangements meeting the specifications.