Permutation Calculator (nPr)

Permutation formula

The number of permutations of r items chosen from n distinct items is nPr = n! ÷ (n − r)!, where n! (n factorial) means n × (n − 1) × … × 1. Example: 10P3 = 10! ÷ 7! = 10 × 9 × 8 = 720.

How to calculate permutations step by step

1. Write down n (how many items you can choose from) and r (how many positions you fill).
2. Start at n and multiply r numbers counting down: n × (n − 1) × … × (n − r + 1).
3. The product is nPr. There is no need to work out the full factorials.

Example: gold, silver and bronze for 8 runners. There are 8 choices for gold, 7 for silver and 6 for bronze: 8 × 7 × 6 = 336 possible podiums.

Types of permutations

In LETTER the six letters include two E's and two T's. Swapping identical letters doesn't create a new word, so the 720 orderings of six letters are divided by 2! for the E's and 2! for the T's: 720 ÷ 4 = 180 distinct arrangements.

Permutation vs combination

Use a permutation when the order of the chosen items matters and a combination when it doesn't. Choosing a president, secretary and treasurer from 10 people is a permutation (10P3 = 720); choosing any 3-person committee is a combination (10C3 = 120). Each committee can be ordered in 3! = 6 ways, which is why nPr = nCr × r!.

How fast permutations grow

Factorials grow extremely quickly: 10! is 3,628,800 and 20! is about 2.43 × 10¹⁸. That is why a shuffled deck of 52 cards (52! ≈ 8.07 × 10⁶⁷ orderings) has almost certainly never been in the same order twice. The calculator switches to scientific notation for very large results.

Where permutations are used

• Passwords and PIN codes – counting possible codes shows how hard they are to guess.
• Scheduling and rankings – orders of speakers, race finishes, tournament seeds.
• Probability – the chance of one specific arrangement is 1 ÷ nPr.
• Genetics and chemistry – arrangements of sequences and molecules.

Permutations in probability

If every arrangement is equally likely, the chance of one particular arrangement is 1 ÷ nPr. The chance of guessing a 4-digit code with no repeated digits in one try is 1 ÷ 5,040 ≈ 0.0198%. Allowing repeated digits raises the possibilities to 10,000 and lowers the chance to 0.01%.

Words with many repeated letters

MISSISSIPPI has 11 letters: one M, four I's, four S's and two P's. The number of distinct arrangements is 11! ÷ (4! × 4! × 2!) = 39,916,800 ÷ 1,152 = 34,650.

Permutations in Excel and Google Sheets

• PERMUT(n, r) returns nPr without repetition – PERMUT(10, 3) = 720.
• PERMUTATIONA(n, r) allows repetition – PERMUTATIONA(10, 4) = 10,000.
• FACT(n) returns n!, and COMBIN(n, r) returns nCr for comparison.

Common mistakes

• Using a permutation when order doesn't matter (a committee, a hand of cards) – that needs combinations.
• Forgetting to divide by the repeats when items are identical.
• Allowing r to exceed n when items can't be reused.
• Treating seats around a round table like seats in a row – rotations of the same seating are not new arrangements.

Permutations with conditions

Many textbook problems add a rule. To seat 6 people in a row with two friends always together, treat the pair as one block: 5 blocks can be arranged in 5! = 120 ways and the pair can swap places, so 120 × 2 = 240 arrangements. The number of seatings where they are NOT together is all seatings minus those: 720 − 240 = 480.

For 'must start with a vowel' or 'must end in an even digit' problems, fill the restricted position first, then arrange the remaining items in the remaining places.