Counting Reference

Permutations and Combinations

Permutations count ordered arrangements. Combinations count unordered groups. Most counting problems become easier once you decide whether changing order creates a new outcome.

Fact Table

Question Type	Order Matters?	Use
Award places, passwords, lineups, schedules	Yes. $ABC$ and $BAC$ are different.	Permutation or slot multiplication.
Committees, teams, hands of cards, selected groups	No. $\{A,B,C\}$ and $\{B,A,C\}$ are the same.	Combination.
All distinct objects arranged in a line	Yes.	$n!$
Objects arranged around a circle	Yes, but rotations repeat.	$(n-1)!$
Repeated identical objects in a word or list	Yes, but identical swaps repeat.	Divide by repeated factorials.

Content Formulas

Permutation

$$\,^nP_r=\frac{n!}{(n-r)!}$$

Combination

$$\,^nC_r=\frac{n!}{r!(n-r)!}$$

Circular Arrangement

$$(n-1)!$$

Repeated Objects

$$\frac{n!}{a!b!c!\cdots}$$

Use permutations when positions or roles are different. Use combinations when the final group is all that matters.

Classic Examples

Award Places

Twelve finalists compete for first, second, and third place. How many results are possible?

Counting MoveFill ordered places. First, second, and third are different roles.

$$\,^{12}P_3=\frac{12!}{(12-3)!}$$ $$\,^{12}P_3=\frac{12!}{9!}$$ $$\,^{12}P_3=12\cdot 11\cdot 10$$ $$\,^{12}P_3=1320$$

Lock Code

A four-digit lock uses digits 0 through 9. Digits may not repeat. How many codes are possible?

Counting MoveUse slot multiplication. Each used digit removes one later choice.

$$N=10\cdot 9\cdot 8\cdot 7$$ $$N=5040$$

Committee

A club chooses 3 representatives from 12 members. There are no officer roles. How many committees are possible?

Counting MoveChoose a group. Divide away the internal order of the selected members.

$$\,^{12}C_3=\frac{12!}{3!(12-3)!}$$ $$\,^{12}C_3=\frac{12!}{3!9!}$$ $$\,^{12}C_3=\frac{12\cdot 11\cdot 10}{3\cdot 2\cdot 1}$$ $$\,^{12}C_3=220$$

Grid Travel

A path crosses a 3-by-3 city grid from the lower-left corner to the upper-right corner, moving only right or up. How many shortest paths are possible?

Counting MoveArrange the required moves. A shortest path has 3 rights and 3 ups.

$$N=\frac{6!}{3!3!}$$ $$N=\,^6C_3$$ $$N=20$$

Stars and Bars

Ten identical candies are shared among four children. A child may receive no candies. How many distributions are possible?

Counting MovePlace 3 dividers among 10 identical candies to create 4 labeled groups.

$$N=\,^{10+4-1}C_{4-1}$$ $$N=\,^{13}C_3$$ $$N=286$$

Circular Seating

Eight students sit around a round table. Rotations of the same seating count as the same arrangement.

Counting MoveFix one person as a reference point, then arrange everyone else.

$$N=(8-1)!$$ $$N=7!$$ $$N=5040$$

Keyring

Eight unique keys go on a ring. The ring can be rotated or flipped over. How many keyrings are distinct?

Counting MoveRemove rotations, then divide by 2 because flipped orders are the same physical ring.

$$N=\frac{(8-1)!}{2}$$ $$N=\frac{7!}{2}$$ $$N=2520$$

Repeated Letters

How many distinct arrangements can be made from the letters in LEVEL?

Counting MoveArrange all letters, then divide by each repeated quantity.

$$N=\frac{5!}{2!2!}$$ $$N=\frac{120}{4}$$ $$N=30$$

Block Method

Six students line up for a photo. Ava and Ben must stand next to each other. How many lineups are possible?

Counting MoveTreat inseparable students as one unit, then arrange the students inside that unit.

$$N=5!\cdot 2!$$ $$N=120\cdot 2$$ $$N=240$$

At Least One Restriction

A 4-person committee is chosen from 6 teachers and 5 students. It must include at least 2 teachers.

Counting MoveCalculate each legal group size separately, then add the legal cases.

$$N=\,^6C_2\,^5C_2+\,^6C_3\,^5C_1+\,^6C_4\,^5C_0$$ $$N=15\cdot 10+20\cdot 5+15\cdot 1$$ $$N=265$$

The legal cases are exactly 2 teachers, exactly 3 teachers, and exactly 4 teachers.