How to Calculate Permutations and Combinations

Permutation P(n, r) or nPr = number of ways to choose r items from n and arrange them. Order matters: ABC and ACB are different.

Combination C(n, r) or nCr (“n choose r”) = number of ways to choose r items from n with no regard to order. {A,B,C} is one set no matter how you list it.

Question type	Use	Example
Rankings, codes, lineups, passwords (position matters)	nPr	1st/2nd/3rd place
Committees, hands of cards, pizza toppings (set only)	nCr	Pick 3 toppings

Rule of thumb: if swapping two chosen people (or digits) creates a new outcome you care about, use permutations. If the outcome is only “who is in the group,” use combinations.

Counts feed probability later: favorable ÷ total. Event probability rules live in the free probability calculator and probability guide.

Factorial n! = n × (n−1) × … × 1, with 0! = 1 by definition.

• 4! = 4×3×2×1 = 24
• 5! = 120
• 6! = 720

n! counts permutations of n distinct items when you use all of them (line up n people: n choices for first slot, n−1 for second, …).

Formulas

nPr = n! / (n − r)! = n × (n−1) × … × (n−r+1)

nCr = n! / (r! × (n − r)!) = nPr / r!

The extra ÷ r! in combinations removes the r! internal orderings of each group that permutations counted separately.

Check relationship

For n=5, r=3: 5P3 = 5×4×3 = 60. 5C3 = 60 / 3! = 60/6 = 10. Same as 5!/(3!2!) = 120/(6×2) = 10.

Constraints: 0 ≤ r ≤ n for standard non-negative integers; otherwise the usual school formulas are undefined or zero in context.

Train yourself with a one-sentence test before picking a formula.

Order matters → permutation

• Race medals: Alice, Bob, Cara vs Cara, Bob, Alice are different podium results
• PIN codes: 1-2-3-4 ≠ 4-3-2-1
• Assigning roles: president vs treasurer from the same two people
• Playlist order if sequence is the product

Order does not matter → combination

• Choosing 4 people for a committee with equal status
• 5-card poker hands (suits and ranks matter as a set, not draw order)
• Selecting 3 pizza toppings from 10 when the pizza is just “has those toppings”
• Picking 2 days off in a week if only the set of days matters

Edge phrasing

“How many ways can 3 of 10 students stand in a line?” → permutation. “How many ways can 3 of 10 students be chosen to present?” without order → combination. “How many ways can 3 of 10 be chosen for president, secretary, treasurer?” → permutation (roles order the selection).

Worked numbers you can reuse on homework and interviews.

Permutations

• Podium: 8 runners, top 3 ordered → 8P3 = 8×7×6 = 336
• Distinct digit codes: 4-digit codes from digits 0–9 with no repeat, order matters → 10P4 = 10×9×8×7 = 5040 (adjust if leading zero is banned)
• Full lineup of 5: 5P5 = 5! = 120

Combinations

• Committee: 12 people, choose 4 → 12C4 = 495
• Cards: 52C5 = 2,598,960 five-card hands
• Toppings: 10 choose 3 → 10C3 = 120

Same n, r — different answers

10 people, pick 3: 10P3 = 720 ordered triples; 10C3 = 120 groups. Factor of 3! = 6 between them, as expected.

When you summarize many trial outcomes (mean score, mean payout), averages use a different toolkit — see the average calculator guide.

Prefer the product form so you never expand full n! for large n.

nPr steps

1. Confirm order matters and 0 ≤ r ≤ n.
2. Write r factors: start at n, multiply downward.
3. Example: 7P3 = 7×6×5 = 210

nCr steps

1. Confirm order does not matter.
2. Compute nPr, then divide by r!, or use nCr = n! / (r!(n−r)!).
3. Cancel factors early: 7C3 = (7×6×5)/(3×2×1) = 210/6 = 35

Calculator workflow

1. Open the free permutation and combination calculator.
2. Enter n (total) and r (chosen).
3. Read both nPr and nCr to compare.
4. Match the formula to the word problem’s order test.
5. State units in words (“ways,” “committees,” “codes”).

For “what percent of hands include an ace?” style follow-ups, convert counts to percents with the free percentage calculator.

Real problems add constraints; formulas assume distinct items and simple choose-r rules unless stated.

• Repetition allowed — PINs with repeated digits use nʳ, not nPr
• Identical objects — divide by factorials of repeats (multiset permutations)
• Circular arrangements — often (n−1)! for n distinct seats around a table
• r > n — standard nPr/nCr without replacement are 0 / undefined for school formulas
• Using nCr when roles differ — undercounts if positions are labeled
• Using nPr when only the set matters — overcounts by r!

Sanity checks: nCn = 1, nC0 = 1, nC1 = n, nPn = n!, nP1 = n, nP0 = 1. 5C2 should equal 5C3 (symmetry: C(n,r)=C(n,n−r)).

Practice three word problems you rephrase in one sentence: “order matters?” If you hesitate, list a tiny case (n=3,r=2) and count by hand before using the formula.