In counting, casework involves splitting the counting problem into two or more mutually exclusive cases and count the number of outcomes in each individual case independently. Then we add the number of outcomes from different cases to yield the total number of outcomes. The following are some situations to use casework:
· Use casework when some cases are different from others.
· Use casework when the problem has multiple steps, and the outcomes of the following steps are different for different outcomes of the first step.
· Use casework when the original counting problem is complex and can be split into smaller disjoint counting problems.
Problem 3‑4
Among 4-digit integers, how many are even numbers whose digits are all distinct?
Solution:
4-digit integers range from 1000 to 9999. For the number to be even, there are
5 choices for the last digit: 
and 
Among
these 5 choices, 
is different from
others. If the last digit is 0, the first digit has 9 choices (1-9); the second
digit has 8 choices; the third digit has 7 choices. If the last digit is 
or 8,
the first digit has only 8 choices as it cannot be 0; the second digit still
has 8 choices; the third digit has 7 choices. Therefore, the total number of
4-digit integers that are even numbers whose digits are all distinct is
General Rule: When we construct integers, since the digit 0 cannot be the first digit in an integer, we often need to treat 0 differently from other digits.
Problem 3‑5
A row in a movie theater has 8 seats. In how many ways can we place 8 people in the row if two of the people, Alex and Jack, refuse to sit together?
Solution: Let’s start with placing Alex and Jack. We have 8 possible choices for Alex: 2 seats at either end of the row and 6 seats not at the ends of the row. Depending on whether Alex is assigned a seat at the end of the row, the number of choices for Jack is different. We have two possible cases:
·
Alex is assigned a seat at the ends of the row. Since Jack cannot
sit beside Alex, we have 6 choices for Jack, and the number of arrangements for
the rest of the 6 people is 
· Alex is assigned a seat not at the ends of the row. Since Jack cannot sit beside Alex, we have 5 choices for Jack, and the number of arrangements for the rest of the 6 people is again 720.
Therefore, the total number of arrangements is 