Assigning 5 Grades to 7 People: How Many Possible Combinations?

[MathWarrior]

Given 5 grades how many different ways are there to assign them to 7 people?

How do you determine that this should be $5^{7}$ and not $7^{5}$?

 

[Mentallic]

How many ways are there to assign 5 grades to the first person? Clearly 5.
For two people, you fix the grade of the first person, and give the 5 grades to the next person, and then change the grade of the first person, and again give the 5 grades to the next person. Do this for all 5 grades of the first person. That's essentially 5*5 = 5² ways to give grades to 2 people.
When you have a 3rd person, similarly you give him a grade and then go through all the possible ways to give a grade to the other two people, which we know is 5² ways, and then repeat this process for all 5 grades on that 3rd person. This is 5*5²=5³.

etc...

 

[Ash L]

Similar to what Mentallic has said. The first person can have 5 different choices of grades, and whatever grades that were assigned to the first person will not affect the number of choices available for the second person.
So, the second person will also have 5 choices of grades. The third person and so on will follow in the same manner.
Hence, you will get 5 x 5 x 5 x 5 x 5 x 5 x 5 = 5^7 ways