Fundamental principle of counting — A/L Combined Mathematics

What must I remember about Fundamental principle of counting?

• Fundamental Counting Principle (Multiplication Rule): If an operation can be performed in m ways, and following this a second independent operation can be performed in n ways, the two operations together can be performed in m * n successive ways.
• Generalized Multiplication Rule: For k sequential tasks where task i can be completed in n_i ways, the total number of composite ways is n1 * n2 * n3 * ... * nk.
• Addition Rule (Sum Rule): If an event A can occur in m mutually exclusive ways, and event B can occur in n mutually exclusive ways, then event 'A OR B' can occur in m + n ways.
• Inclusion-Exclusion Principle for two events: |A ∪ B| = |A| + |B| - |A ∩ B|.
• Tree diagrams: Graphical branching structures representing sequential multi-stage decision processes and counting paths.

Test yourself (1 minute)

1. Restrictions must be filled FIRST

   In constrained counting problems (e.g., forming even numbers, digits > 5000), always fill the constrained positions (unit digit, first digit) before filling unrestricted positions.

2. With vs Without Repetition

   Forming numbers/words with repetition allowed gives n^k possibilities; without repetition gives n*(n-1)*...*(n - k + 1) = nPk possibilities.

3. Pigeonhole Principle

   If n items are put into m containers and n > m, at least one container must contain more than one item.

Practise Fundamental principle of counting questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 23

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Year 2 · Term 1