Permutations and Combinations

Counting principles

• Addition principle: if a task can be done in one of several separate ways (case 1 or case 2), add the counts.
• Multiplication principle: if a task is done in stages (step 1 and then step 2), multiply the counts.

Permutations: order matters

The number of ways to arrange $r$ objects chosen from $n$ distinct objects:

$$P^n_r = \frac{n!}{(n - r)!}$$

Arranging all $n$ distinct objects in a row: $n!$ ways.

Combinations: order does not matter

The number of ways to choose $r$ objects from $n$ distinct objects:

$$C^n_r = \frac{n!}{r!(n - r)!}$$

Quick check: choosing a committee or a hand of cards is a combination. Arranging people in seats, forming numbers or passwords is a permutation.

Common conditions

Condition	Method
Certain items must be together	Tie them into one block, arrange the blocks, then multiply by the arrangements inside the block
Certain items must not be next to each other	Total − (number with them together), or arrange the others first and insert into the gaps
A certain position is restricted (e.g. even number)	Fill the restricted position first
"At least one"	Total − (number with none)

Example: 5 boys and 3 girls in a row with the 3 girls together: $6! \times 3! = 4320$.

Common mistake: counting "at least one girl" by adding cases and double-counting. Using "total − none" is quicker and safer.