Counting Principles

The Product Rule states that if one event can occur in $m$ ways and a second event can occur in $n$ ways, then the sequence of two events can occur in $m \times n$ ways.

This principle applies when events are independent and occur in succession (the 'AND' rule).

It can be extended to any number of events: if there are $k$ events with $n_1, n_2, ..., n_k$ ways to occur, the total number of outcomes is $n_1 \times n_2 \times ... \times n_k$.

Example: If a menu offers 3 starters and 5 main courses, there are $3 \times 5 = 15$ possible two-course meals.

To arrange $n$ distinct objects in a row, there are $n$ choices for the first position, $n-1$ for the second, and so on, until 1 choice remains for the last position.

This is represented by the factorial notation: $n! = n \times (n-1) \times (n-2) \times ... \times 1$.

If repetitions are allowed for $n$ objects in $r$ positions, the total number of ways is $n^r$.

Key Formula: The number of ways to arrange $n$ distinct objects is $n!$.

Permutations are used when the position is unique (e.g., Gold, Silver, Bronze medals).

Combinations are used when the items are treated as a single set (e.g., choosing 3 players for a team).

Identify the Operation: Read the problem carefully for 'AND' (multiply) versus 'OR' (add) logic.

Check for Repetition: Determine if an item can be used more than once. If repetition is allowed, factorials and standard permutation formulas do not apply; use the product rule with constant options.

Constraint First: Always deal with specific constraints (e.g., 'the first digit cannot be zero') before calculating the remaining positions.

Sanity Check: If the calculated number is extremely large or small, re-evaluate if you should have used a combination (smaller result) instead of a permutation (larger result).