The Counting Principle

Everything in permutations and combinations reduces to one principle: if a task happens in stages with $m$ choices then $n$ choices, the total is $m \times n$. Factorials, ${}^nP_r$ and $\binom{n}{r}$ are this principle pre-packaged, and when a question doesn’t fit a package, the principle itself, written as slots, always works.

The slot method

How many 4-digit numbers can be formed from the digits 1–7 without repetition? Draw four slots; fill in the choices: $7 \times 6 \times 5 \times 4 = \mathbf{840}$

The written slot product is both the thinking tool and the visible method, an examiner can award a slip-damaged answer that shows correct slots, and nothing at all to a bare wrong number.

Restrictions: most constrained slot first

How many of those 840 are even? Fill the constrained slot first, the last digit must be 2, 4 or 6: 3 choices. Remaining three slots from the six unused digits: $6 \times 5 \times 4 = 120$. Total: $120 \times 3 = \mathbf{360}$

Constraint-first ordering isn’t style; it’s correctness. Filling free slots first makes the constrained slot’s count depend on earlier choices, the count fractures into cases, and most students who try it double-count. Standard constraint types: fixed ends (“must start with a vowel”), parity (“even numbers”), forbidden positions, all yield to constrained-slot-first.

Multiply or add? The and/or test

Stages done in sequence (and) multiply. Mutually exclusive alternatives (or) add:

Digits from 1–7, 4-digit numbers greater than 5000: first digit 5, 6 or 7 (3 ways), then $6 \times 5 \times 4$ for the rest → $3 \times 120 = 360$. One case. But “odd numbers greater than 5000” splits: first digit 5 or 7 overlaps the last-digit-odd constraint → handle as cases and add (first digit odd vs even), because the constraints interact.

The diagnostic before any arithmetic: do my constraints touch the same slots? If yes, split into cases, count each with slots, add.

Common mistakes

• Free slots filled before constrained ones
• Cases that should add being multiplied (and vice versa)
• Repetition allowed/disallowed misread from the question
• Interacting constraints handled as if independent
• The slot product unwritten, method marks abandoned

Full topic context: P&C notes · the packages: factorials, ${}^nP_r$, $\binom{n}{r}$.