Ten huskies are to be selected to pull a dogsled. How many different arrangements of the 10 huskies on a dogsled are possible?

Since there are 10 dogs, and we are assuming that all 10 will be participating, you can use the factorial 10! which means 10*9*8*7*6*5*4*3*2*1.

The reasoning behind this is that you have 10 choices for the first position. To write this out, assume that you have dogs
A B C D E F G H I J
For the first dog chosen, it can be A, B, C, D, E, F, G, H, I, or J, so 10 dogs. Lets say we choose dog G to go first. This means we have 9 dogs left, A, B, C, D, E, F, H, I, and J. So we have 9 for the second spot. Each time we will be reducing the number of available dogs by 1, which gives us the 10*9*8*7*6*5*4*3*2*1. For a final answer of 3,628,800.

There are 10 dogs and 10 positions to fill. Select one dog to fill the first position. You have 10 choices for the first position. Choose a dog for the second position. How many dogs can you choose from now? Only 9, since 1 went into the first position. The fundamental counting principle tells us that the number of choices in each category times each other give the total number of choices there are. So we will multiply 10 times 9 to give the number of  choices for the first 2 positions. But there are more positions! For the third position there are 8 choices and for the fourth position there are 7 choices. All the way down to the last position there will only be one dog left. Using the Fundamental Counting Principle, the total number of choices will be 10*9*8*7*6*5*4*3*2*1, which gives us 3,628,800. This is also known as "ten factorial", which is written 10!.

The question is a little vague.
Ten huskies are to be selected to pull a dogsled. The number of ways in which this can be done is dependent on the total number of huskies from which these ten are being selected. If the total number were N, the number of ways of selecting the ten huskies would be given by NC10 = (N!)/(10!*(N-10)!)
As we do not have the value of N, we move over to the next part of the question. Ten huskies have been selected to pull the dogsled and now the number of different arrangements of the ten huskies has to be determined.
There are 10 positions in which the huskies can stand. The first position can be filled by any of the 10. Once this is done, the second position can be filled by any of the remaining 9. The next position can be filled by any of the remaining eight. Moving ahead in the same way, by the time we reach the tenth position there is only one husky remaining.
The number of ways of arranging r objects from a set of n is given by nPr which is equal to (n!)/((n-r)!). Here, n = 10 and r is also equal to 10.
10P10 = (10!)/((10-10!))
= (10!)/(0!)
The value of 0! is 1.
10P10 = 10! = 10*9*8*7...*1= 3,628,800
The total number of different arrangements of the 10 huskies on a dogsled is equal to 3,628,800.