The multiplication principle can be applied here. There are 5 ways to get directly from Point A to Point B, 3 ways to get directly from Point B to Point C, and 2 ways to get directly from Point C to Point A. Since traveling the same path is not permitted, there are 4 ways left to get directly from Point A to Point B, 2 ways left to get directly from Point B to Point C, and 1 way left to get directly from Point C to Point A. Multiply these to get

 possible routes.

The multiplication principle can be applied here. There are 5 ways to get from Point A to Point B, 3 ways to get from Point B to Point C, 3 ways to get from Point C to Point B, and 5 ways to get from Point B to Point A. Multiply:

 possible routes.

The multiplication principle can be applied here. There are 5 ways to get from Point A to Point B, and 3 ways to get from Point B to Point C. Since traveling the same path twice is not allowed,  there are 2 ways left to get from Point C to Point B, and 4 ways left to get from Point B to Point A.

Multiply:

There are 120 possible routes.

Since the table is circular, you need to find the total number of orders and divide this number by 7.

The total number of different orders that the placemats could be set in is 7! (7 factorial).

7!/7 = 6! = 720

Note that had this been a linear, and not circular, arrangement there would be no need to divide by 7. But in a circular arrangement there are no "ends" so you must divide by N! by N to account for the circular arrangement.

To start, the first player makes 6 high-fives. The second player needs to make only 5 new high-fives because he doesn't need to re high-five the first player. The third player needs to only make 4 high-fives and so on.

This adds up to 6 + 5 + 4 + 3 + 2 + 1 = 21

There are four pairs of seats directly across from each other: 1-6, 2-5, 3-8, 4-7.

There are four married couples to be placed in these four pairs of seats; since order is relevant, these are permutations. The number of ways to seat these couples is equal to the number of orderings, or permutations, of four elements from a set of four:

,

which is equal to 

Within each pairing, there are two ways for the husband to select one seat and the wife to select the other, and there are four such pairings, so, applying the multiplication principle, the number of arrangements is 

Since Seats 1 and 8 will be occupied by the Williamses, there are two ways to fill these seats - Mr. Williams in Seat 1 and Mrs. Williams in Seat 8, and the opposite case.

The remaining six persons will be seated in the other six seats; since order is material here, this is a number of permutations. The number of permutations of six elements from a set of six is

,

which is equal to

.

By the multiplication principle, the number of seating arrangements is

.

One way to look at this is as follows:

Yvonne can sit at any one of the eight seats—eight ways to make this decision.

Yvonne can fill the remaining seven seats with seven out of twenty classmates. Since order is important, this is a permutation of seven elements chosen from a set of twenty; there are  ways to make this decision.

By the multiplication principle, there are  ways Yvonne can invite seven classmates and seat them and herself at the table.

Kacey works 5 hours a day for 3 days and 9 hours a day for 2 days. (5)(3) = 15 and (2)(9) = 18. 15 + 18 = 33 hours worked a week. $363/33 hours = $11/hr.

Since the remainder when k is divided by 2 is 1, this means that k is not an even number. Therefore, we can immediately eliminate 30 from our possible answer choices.

Because the remainder when k is divded by 3 is 0, we know that k must be a multiple of 3. This means we can eliminate 23 from our answer choices.

The only choices left are 63, 27, and 57. When 27 and 57 are divded by five, the remainder is two. Thus, only 63 works, because when 63 is divded by five, the remainder is three.

The answer is 63.