Combination Formula Online: Choose Without Order

Decide whether order changes an outcome

A combination selects r distinct items from n distinct items without caring about their order. Choosing Ana, Ben and Chen for an unranked committee is the same group regardless of the order in which their names are written. Assigning those people to chair, secretary and treasurer is different because the roles are distinct. The first task uses combinations; the second uses permutations. Establish the outcome definition before selecting a formula, because both calculations can return plausible-looking whole numbers.

Apply the factorial formula

The number of combinations is C(n,r) = n!/[r!(n−r)!], where 0! = 1. For choosing three people from eight, C(8,3) = 8!/(3!5!) = (8×7×6)/(3×2×1) = 56. The cancellation avoids writing every factorial term. Enter n = 8 and r = 3 in the calculator and read the line beginning C. The separate permutation line is P(8,3) = 336, which counts the six possible orders of each selected three-person group.

Understand the division by r factorial

An ordered selection of r distinct items has n(n−1)...(n−r+1) possibilities. Each unranked group of r items appears r! times among those ordered selections. Dividing by r! removes that repeated counting. For the three-person example, each group can be arranged in 3! = 6 orders, so 336/6 = 56. This reasoning assumes the items are distinct and selected without replacement; it cannot be transferred unchanged to repeated identical objects or a process that allows the same item to be chosen again.

Use symmetry and edge cases as checks

Choosing three of eight is equivalent to identifying the five left out, so C(8,3) = C(8,5) = 56. In general, C(n,r) = C(n,n−r). Choosing none or all gives one possible subset, so C(n,0) = C(n,n) = 1. Choosing one gives C(n,1) = n. A value r greater than n cannot select that many distinct items without replacement, and the local form rejects it. Fractional or negative item counts are also outside this basic counting model.

Work a constrained selection in separate stages

Suppose a committee needs two people from a group of five and one person from a separate group of four. Choose the first pair in C(5,2) = 10 ways and the final person in C(4,1) = 4 ways, giving 10×4 = 40 possible committees. Using C(9,3) would ignore the required composition and count additional groups. The local form evaluates one n-choose-r calculation at a time; the multiplication and constraint interpretation belong to the overall problem setup.

Identify selection with repetition

Choosing three scoops from four flavours when repeated flavours are allowed and scoop order is ignored has a different count. The standard combinations-with-repetition formula is C(n+r−1,r), giving C(6,3) = 20. This is a manual model transformation before using the ordinary combination form. If scoop order matters, there are instead 4^3 = 64 ordered sequences. The same physical words can describe different outcome spaces, so state replacement and ordering rules explicitly.

Keep counting separate from probability

A count of 56 possible committees does not establish that each is equally likely to be selected. Dividing one by 56 gives a particular committee's probability only under a uniform selection process. Additional eligibility rules, unequal selection weights or dependent stages change that probability model. The calculator's exact integer count also should be copied without converting it to a limited-precision spreadsheet number when every digit matters. Large combination counts can exceed ordinary numeric precision even though the local output remains an exact integer.

Check the smaller selection size

For C(100,98), symmetry gives C(100,2) = 100×99/2 = 4950. This shorter calculation independently checks the exact output without constructing large factorials. It also confirms that selecting almost all items can have the same count as selecting only the few that are left out.

Related guides

Calculate a Percentage: Part, Whole and Percent Of

Choose the right percentage equation, calculate a part of a whole, recover an unknown whole, and avoid confusing percent with percentage change.

Basic Calculations: Order, Signs and Checks

Work basic arithmetic expressions with parentheses, fractions, powers and negative numbers, and check results with estimates and inverse operations.

Calculate an Average and Check the Denominator

Calculate arithmetic means from raw numbers, distinguish zero from missing data, and handle weighted groups, outliers and average rates correctly.