Unordered colored tiles beside a neatly ordered line of the same tiles. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Decide whether swapping positions creates a new outcome
Choosing three committee members from ten distinct people ignores order: the same three people form the same committee regardless of how their names are written. Assigning chair, secretary and treasurer to three people is ordered: swapping two roles changes the outcome. Both questions choose three people, but they count different objects.
Compare the counts
There are 10×9×8 = 720 ordered assignments to three distinct roles. Each unordered group of three appears in 3! = 6 orders. Dividing gives C(10,3) = 120 committees. The Combination Calculator returns the unordered count, so its result for n = 10 and r = 3 is 120, not 720.
Check what “distinct” means
The objects in an ordinary subset calculation are distinguishable. If people have eligibility constraints, roles require different qualifications or some combinations are forbidden, C(n,r) alone does not enforce those restrictions. Count the permitted outcome space explicitly. A zero result for choosing more distinct objects than exist means no such subset is possible; choosing zero objects gives one empty subset.
Counting is not automatically probability
A ratio of favorable subsets to total subsets gives a probability only when the selection mechanism makes those subsets equally likely, or when their unequal weights are otherwise accounted for. Choosing employees with different selection weights is not the same as uniform committee sampling. Likewise, allowing repeated types turns subsets into multisets and changes the count. Read the repetition guide before enabling that option. The coin-count calculator uses combinations to count placements of heads under its fixed-p independent-flip model, which is a different use from assigning committee roles.
Check the model behind your result
Use the Combination Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26