Combination Formula Calculator

Combination Calculator Form

Calculation mode

Standard nCr or stars-and-bars style counting.

n: total distinct items

Examples: students, cards, products, or symbols.

r: items selected

Order never matters in combination counting.

Scientific precision

Controls decimal places in scientific notation.

Optional comparison r

Compare your current result with another selection size.

Optional chart limit

Set the largest plotted k value for the graph.

Example Data Table

Scenario	n	r	Mode	Formula Applied	Result
Choose 2 tools from 5	5	2	Without repetition	C(5,2)	10
Choose 3 students from 10	10	3	Without repetition	C(10,3)	120
Choose 10 numbers from 20	20	10	Without repetition	C(20,10)	184,756
Choose 3 scoops from 4 flavors	4	3	With repetition	C(4+3-1,3)	20
Choose 5 balls from 6 bins	6	5	With repetition	C(6+5-1,5)	252

Formula Used

1) Standard combination formula

C(n, r) = n! / (r! × (n − r)!)

Use this when order does not matter and each item can only be chosen once. The calculator also applies symmetry, so C(n, r) = C(n, n − r).

2) Combination with repetition

C(n + r − 1, r) = (n + r − 1)! / (r! × (n − 1)!)

Use this when order still does not matter, but the same item type may be selected more than once. This is the classic stars-and-bars counting model.

How to Use This Calculator

Select whether repetition is allowed.
Enter n, the number of distinct items.
Enter r, the number of items selected.
Set the scientific precision you want.
Optionally enter another r value for comparison.
Optionally set a chart limit for the Plotly graph.
Click Calculate Combinations.
Review the result, chart, and export buttons above the form.

Frequently Asked Questions

1) What does a combination measure?

A combination counts how many unordered groups can be formed from a larger set. It ignores arrangement, so selecting A then B is the same as selecting B then A.

2) When should I use combinations instead of permutations?

Use combinations when order does not matter. Use permutations when order matters. Team selection, menu bundles, and lottery picks often use combinations.

3) What is the difference between repetition and no repetition?

Without repetition, each item can appear only once. With repetition, the same item type can be chosen again. The two cases use different formulas.

4) Why does the chart use log10 values?

Combination counts can grow extremely fast. A log10 scale keeps the graph readable and makes growth trends easier to compare across small and large results.

5) Why do C(n, r) and C(n, n − r) match?

Choosing r items automatically determines which n − r items were not chosen. Those two views count the same collection of outcomes, so the totals are identical.

6) Can this calculator handle very large inputs?

Yes, it can estimate very large values with scientific notation. Exact whole-number output is shown for moderate sizes to keep the page fast and practical.

7) What does “1 in result” mean?

It shows the probability of one specific unordered selection if every valid combination is equally likely. For example, 1 in 120 means one exact group among 120 possible groups.

8) Where is this calculator useful?

It helps in probability, statistics, coding, scheduling, inventory grouping, classroom selection, genetics, card analysis, and any task involving unordered choices.