Counting methods make probability efficient when listing all outcomes is impractical.

Factorial

\[ n!=n(n-1)(n-2)\cdots 1,\quad 0!=1. \]

Permutations (order matters)

\[ {}_nP_r=\frac{n!}{(n-r)!}. \]

Use for arrangements where position/order is important.

Combinations (order not important)

\[ {}_nC_r=\frac{n!}{r!(n-r)!}. \]

Use for selections/groups.

Probability with counting

\[ P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}. \]

With/without replacement

• With replacement: probabilities stay same each draw.
• Without replacement: probabilities change each draw.

Exam strategy

• First decide: arrangement or selection?
• Check if repetitions are allowed.
• Use structured counting (tree/cases) for restrictions.
• Simplify probability fractions at end.