Cambridge O Level Additional Mathematics · Syllabus 4037 · Permutations and Combinations
Mutually Exclusive Cases
What is Mutually Exclusive Cases?
A way of splitting a counting problem into separate situations such that every outcome belongs to exactly one of them, so that their counts may be added without double counting. Cases are mutually exclusive when no outcome can satisfy two of them at once, which is guaranteed if they are defined by the exact value of some quantity, such as the exact number of girls on a team. Within a single case the successive independent choices are multiplied, because they are stages of building one outcome, while the totals for separate cases are added, because the cases are alternatives. Overlapping cases inflate the answer invisibly, since the excess never appears as an obviously impossible value.
This definition is part of the Permutations and Combinations chapter in Cambridge O Level Additional Mathematics.
Mutually Exclusive Cases in context
Every permutation can be built by first selecting the r objects with \({}^nC_r\) and then arranging them in \(r!\) ways, so \({}^nP_r=r!\times{}^nC_r\); this bridge lets you check one calculation against the other rather than mixing the two formulas. In restriction and case problems, "and then" multiplies independent choices, "or, never both" adds mutually exclusive cases, and "at least one" is usually found by subtracting an unwanted count from the total. Also note \(0!=1\), since there is exactly one way to arrange nothing, which is what keeps both formulas working correctly at their end points.
Questions students ask about Mutually Exclusive Cases
How do you handle "and", "or" and "at least one" in a counting problem?
"And then" multiplies independent, consecutive choices, because every outcome is built by making all of them together. "Or, never both" adds mutually exclusive cases, defined by an exact number so they cannot overlap, since each outcome then belongs to exactly one case. "At least one" is usually found by subtracting an unwanted count from the unrestricted total, not by adding cases that could overlap and double-count the same outcome.