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Count choices made one after another with the multiplication principle, separate cases with the addition principle, and meet the factorial.
Arrangements when order matters: the formula for nPr, arrangements with repeated objects, and arrangements under conditions such as letters that must stay together.
Selections when order does not matter: the formula for nCr, its properties, and problems that mix selecting and arranging.
The counting problems that JEE, A-Level Further Maths and maths olympiads set most: circular arrangements, distributing objects into groups, the gaps method, and counting by complements and cases, with a full timed paper.