Permutations vs Combinations: nPr and nCr Formulas, Factorials, and Examples

Ask "how many ways are there to pick 3 people from 5?" and the answer might be 10, or it might be 60. Same "3 from 5," so why does the number change? The key is whether order matters. When order matters it is a permutation; when it does not, it is a combination. This article sorts out the two using formulas and concrete examples, and gently covers the points where people often stumble.

Permutations vs Combinations

When counting outcomes, the most important question is: are you arranging things, or just choosing them?

• Permutation: you arrange the items you choose. A different order counts as different. The symbol is nPr.
• Combination: you only choose, without arranging. A different order counts as the same. The symbol is nCr.

For example, choosing 2 from "A, B, C": as combinations there are 3 ways, {A,B}, {A,C}, {B,C}. But as permutations, choosing A and B counts "AB" and "BA" separately, giving 6 ways: AB, BA, AC, CA, BC, CB. As a rule, permutations yield a larger number than combinations, because after choosing, each selection branches further by the number of arrangements.

A handy test is to ask whether the order carries meaning, as with seating, rankings, or passwords. If it does, use permutations; if not, use combinations. To check with real numbers, try changing n and r in the Combination Calculator and comparing the results side by side.

The nPr Formula and Factorials

To understand the permutation formula, you first need the factorial. A factorial is written with "!" and means multiplying every integer from 1 up to that number. For example, 5! = 5×4×3×2×1 = 120. That equals the total number of ways to arrange 5 items in a row: 5 choices for the first spot, 4 for the next, 3 for the one after, and so on.

So how do we count permutations of r items taken from n? The first pick has n choices, the second has n-1, the third has n-2, and the r-th has n-r+1. Multiplying these gives the number of permutations.

nPr = n × (n-1) × (n-2) × … × (n-r+1) = n! ÷ (n-r)!

For instance, "pick 3 from 5 and line them up" gives 5P3 = 5×4×3 = 60. Written with factorials, 5! ÷ 2! = 120 ÷ 2 = 60, the same answer. When r = n, nPn = n!, which matches simply arranging everything.

Factorials grow huge very quickly: even 10! is already about 3.6 million. For large numbers that are painful by hand, the factorial (x!) function in the Scientific Calculator lets you compute safely.

The nCr Formula

Because combinations do not arrange, they produce a smaller number than permutations. Specifically, nPr already contains duplicate "rearrangements of the same members," so dividing by that duplication gives the combination. There are r! arrangements of r items, so the formula is:

nCr = nPr ÷ r! = n! ÷ ( r! × (n-r)! )

Counting "3 from 5" as combinations gives 5C3 = 60 ÷ 3! = 60 ÷ 6 = 10. The 60 permutations divided by the 3! = 6 rearrangements of 3 people equals 10.

A useful property is the symmetry nCr = nC(n-r). Choosing 7 from 10 is the same as choosing the 3 to leave out, so 10C7 = 10C3 = 120. When you are choosing many, compute with the smaller leftover count instead. Also note nC0 = 1 and nCn = 1 (there is exactly one way to choose none or to choose all). When numbers get large and mental math is hard, just enter n and r into the Combination Calculator for an instant answer.

Examples: Lotteries and Team Splits

The difference is clearest in everyday situations.

1. Lottery prizes (ranked = permutation)

Awarding 1st, 2nd, and 3rd prize one each from 10 tickets means the prizes differ in weight, so order matters. That is a permutation: 10P3 = 10×9×8 = 720. Because we distinguish "who gets 1st," the count is large.

2. Choosing representatives (unranked = combination)

By contrast, just "pick 3 committee members from 10" has no ranking among the three, so only being chosen matters. That is a combination: 10C3 = 120, which equals the 720 permutations divided by the 6 rearrangements of 3 people.

3. Team splits (mind the duplication)

Team splitting needs care. Splitting 8 people into Team A of 4 and Team B of 4: choose the 4 for A and the rest go to B automatically, so 8C4 = 70. But splitting 8 into two indistinguishable groups of 4 means there is no A/B distinction, so divide further by 2!, giving 70 ÷ 2 = 35. This is a classic case where whether the groups have labels changes the answer. If you are unsure about such distinctions, build the count piece by piece while checking each value in the Combination Calculator to avoid mistakes.

Counting When Repetition Is Allowed

So far we assumed "you cannot choose the same item twice." But sometimes repetition is allowed, like rolling a die repeatedly or picking any number of the same snack. The formulas then change.

A permutation with repetition lets you pick from all options every time, so making one of n choices r times gives n^r (n to the power r). For example, a 4-digit PIN has 10 options (0-9) per digit over 4 digits, so 10^4 = 10000.

A combination with repetition counts "order ignored, any number of the same item allowed," with the formula nHr = (n+r-1)Cr. For example, "pick 5 drinks total from 3 kinds (any number of each)" gives 3H5 = (3+5-1)C5 = 7C5 = 21. It is easier to grasp by recasting it as arranging "dividers and items," which ultimately reduces to an ordinary combination. Since the numbers can get large, it is reassuring to double-check powers and factorials with the Scientific Calculator.

The Link to Pascal's Triangle

The combination value nCr is deeply tied to a beautiful arrangement of numbers called Pascal's triangle. Each row starts and ends with 1, and every inner number is the "sum of the two numbers directly above it."

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

The numbers in row n (counting the top as row 0) are exactly nC0, nC1, …, nCn from left to right. For example, row 4's "1 4 6 4 1" is 4C0=1, 4C1=4, 4C2=6, 4C3=4, 4C4=1.

The "sum of the two above" rule corresponds in symbols to nCr = (n-1)C(r-1) + (n-1)Cr (Pascal's rule). You can grasp it intuitively by focusing on one specific person and splitting into the cases where that person is chosen or not. The triangle also shows other properties directly: each row sums to 2^n, and it is left-right symmetric (the nCr = nC(n-r) above). Understanding this structure, rather than memorizing formulas, makes applications easier. Check the actual numbers with the Scientific Calculator or the Combination Calculator and compare them against the triangle.

Please note that this article is intended to provide general learning information. When using it for exams or assignments, the scope and the notation for symbols (how nPr and nCr are written) can differ by textbook, so confirm against your latest textbook and official materials before relying on it.