How it works
In casework you split the possibilities into separate cases, count each one and add. It works when the cases are disjoint (no outcome is in two cases) and complete (every outcome is in some case).
Good cases usually come from fixing one quantity: the first digit, the largest die, the number of girls on a committee. Make an organised list or table so you can see that nothing is missing, and check your total another way when you can. If the number of cases gets large, that is a sign that complementary counting or a formula might be faster.
Casework Counting practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Two dice are rolled. In how many ways can the sum be 4?
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Answer: 3
The ordered pairs are (1, 3), (2, 2) and (3, 1): 3 ways.
Roll two dice. How many outcomes have sum 9 or 11?
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Answer: 6
Sum 9: (3,6), (4,5), (5,4), (6,3), which is 4 outcomes. Sum 11: (5,6), (6,5), which is 2. The cases don't overlap, so 4 + 2 = 6.
How many numbers from 100 to 999 have digit sum 5?
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Answer: 15
Fix the hundreds digit h; the other two digits must add to 5 − h. For h = 1, 2, 3, 4, 5 that gives 5, 4, 3, 2, 1 ways. Total 15.
Three dice are rolled. In how many ways can the sum be 6?
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Answer: 10
Cases by the set of values: {4, 1, 1} has 3 orders, {3, 2, 1} has 6 orders and {2, 2, 2} has 1. Total 3 + 6 + 1 = 10.
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Related topics
Related practice sets in the app
All Counting & Probability topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice