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Five people get into an elevator. There are seven floors on which they can get off. What is the total number of different ways they can do this?

Let's begin by figuring out first the total number of ways they can do this. Each person can choose from 7 floors. There are 5 people. So, the total number is 7x7x7x7x7 = 16,807.

Now the fun part. We have think of the different cases. They are:
case 1: everyone on a different floor;
case 2: two people on the same floor and everyone else on different floors (4 floors altogether);
case 3: two people on one floor, two other people on another floor, and the last person on yet another floor (3 floors altogether);
case 4: three people on one floor and the other two on two different floors (3 floors altogether);
case 5: three people on one floor and the other two on another floor (2 floors altogether);
case 6: four people on one floor and the last person on another floor (2 floors altogether);
case 7: everyone on the same floor.

Is your brain hurting yet?

Let's tackle these cases one at a time. The method I use I call the choose/arrange method. I call it that because what I do in a problem like this is first choose the floors, and then arrange the floors among the 5 people.
The way I choose the floors is according to how many people choose each floor. You'll see what I mean once we start solving the problem.

Case 1: There are 7 floors to choose from. There is only 1 person per floor. So, the total number of ways I can choose the 5 floors is
The number of ways I can arrange the 5 floors is 5! = 120. So, the total number of way in which everyone chooses a different floor is 21 x 120 = 2,520.

Case 2: There is 1 floor which 2 people are choosing. There are choices for that floor. There are now six floors left now for the other 3 people. There are ways in which this can be done. So, the total number of floor choices for this case is 7 x 20 = 140.
To make my life easier (something I like to do once in a while), I will give each chosen floor a letter. In this case, we have AABCD because 2 people chose 1 floor and everyone else chose a different floor. Now I can use basic counting rules to arrange the letters in AABCD. The number of way I can do that is 5!/2! = 60. So, the total number for this case is 140 x 60 = 8,400.

Case 3: There are 2 different floors in which 2 people choose that floor. There are ways in which those floors can be chosen. That leaves 5 floors for the last person to choose. The total number of floor choices for this case is 21 x 5 = 105.
My floor letter pattern for this case is AABBC. The number of ways in which the letters in this word can be arranged is 5!/(2!2!) = 30. So, the total number for this case is 105 x 30 = 3,150.

Case 4: We have 3 people choosing the same floor. I'm sure you figured out that there are 7 choices for that floor. That leaves 6 floors to choose from for the other 2 people, and So the total number of floor choices for this case is 7 x 15 = 105.
The floor lettern pattern for this case is AAABC. The number of ways in which the letters in this word can be arranged is 5!/3! = 20. So, the total number for this case is 105 x 20 = 2,100.

IS IT GETTING EASIER YET???

Case 5: Once again, we have 3 people choosing the same floor. Once again, we have 7 choices for that floor. That leaves 6 floors to choose from for the other 2 people. So the total number of floor choices for this case is 7 x 6 = 42.
The floor lettern pattern for this case is AAABB. The number of ways in which the letters in this word can be arranged is 5!/(3!2!) = 10. So, the total number for this case is 42 x 10 = 420.

Case 6: Four people choose 1 floor out of 7. The last person chooses 1 floor out of the remaining 6. Have we been here before? The total number of floor choices for this case is 7 x 6 = 42.
The floor lettern pattern for this case is AAAAB. The number of ways in which the letters in this word can be arranged is 5!/4! = 5. So, the total number for this case is 42 x 5 = 210.

Case 7: Everyone chooses the same floor. Number of floor choices: 7. Floor pattern: AAAAA. Number of ways this can be arranged: 1. Total for this case: 7

If you add up all the numbers from all the cases, you will make the brilliant observation that they add up to 16,807.