How many different 5-person committees can you pick from 12 people? The Combination Calculator counts groups where order does not matter. You enter the total and the group size, and it returns two counts.
It suits students, teachers, and anyone sizing up card hands, lottery picks, or team choices. The tool uses big-integer math, so large answers keep every digit.
Table of Contents
- What the Calculator Does
- How to Use It, Step by Step
- The Combination Formulas
- Worked Example
- How to Read the Results
- Common Uses and Mistakes
- Limitations and Assumptions
- FAQs About the Combination Calculator
What the Calculator Does
The tool has two inputs, two results, and one button.
- Total Objects (n): how many different items you choose from. It starts at 12.
- Sample Size (r): how many items you choose. It starts at 5.
- Combinations nCr (Order Irrelevant): the number of groups with no item used twice.
- Combinations with Repetition (Order Does Not Matter): the number of groups when an item may be picked again.
- Calculate Combinations runs the math. The results also update as you type and when you press Enter.
How to Use It, Step by Step
- Enter a whole number from 0 to 100 in Total Objects (n).
- Enter a whole number from 0 to 100 in Sample Size (r). For nCr it must not be larger than n.
- Press Calculate Combinations, or press Enter in either box.
- Read the two cards under "Unordered Selection Result".
If a box holds something the tool cannot use, a red message appears under the inputs. The cards then show N/A instead of old numbers.
The Combination Formulas
The tool builds nCr by multiplying and dividing in steps, which keeps every answer exact. A factorial, written fact(n) here, is the product of all whole numbers from 1 to n.
fact(n)÷fact(r) × fact(n - r)=nCr
- nCr = fact(n) ÷ (fact(r) × fact(n - r)).
- With repetition = fact(n + r - 1) ÷ (fact(r) × fact(n - 1)).
- By definition, fact(0) equals 1, so nC0 and nCn both equal 1.
- The tool uses the smaller of r and n - r as the number of steps.
With n = 0, the with-repetition card shows 1 when r is 0. It shows 0 when r is above 0.
Worked Example
Andre runs a pizza shop with 12 toppings. A customer picks 5 toppings, and order does not matter.
| Result | Calculation | Output |
|---|
| nCr, n = 12, r = 5 | fact(12) ÷ (fact(5) × fact(7)), or 95,040 ÷ 120 | 792 |
| With repetition | fact(16) ÷ (fact(5) × fact(11)), or C(16, 5) | 4,368 |
The first line uses the shortcut 12 × 11 × 10 × 9 × 8, which is 95,040. Dividing by fact(5), which is 120, gives 792.
Another example is a 6-number lottery from 49 numbers. The tool shows 13,983,816 for nCr and 25,827,165 with repetition.
How to Read the Results
Combinations nCr counts each group once, no matter how its members are listed. Choosing Ana, Ben, and Cy is the same as choosing Cy, Ana, and Ben.
Combinations with Repetition (Order Does Not Matter) allows the same item more than once, such as three scoops of the same flavor. When r is not larger than n, it is at least as large as nCr.
The nCr value is symmetric. Choosing 5 from 12 gives the same count as leaving out 7 from 12.
The tool adds comma separators itself, so numbers look the same on every device. For example, 52 choose 5 gives 2,598,960, the number of five-card poker hands.
Common Uses and Mistakes
Use a combination when only membership matters, such as a hand of cards or a jury. If order matters, use the permutation calculator instead. It gives a count fact(r) times larger.
To turn a count into a chance, divide the good groups by all groups. The probability calculator can then combine two such chances.
For repeated yes or no trials, try the binomial probability calculator. It uses nCr inside its formula.
The scientific calculator can check a small answer by hand. To reduce a ratio of two counts, use the fraction calculator. More tools live in the math and science calculators hub.
Limitations and Assumptions
- Only whole numbers from 0 to 100 are accepted. Larger values, such as 100000, are rejected at once with a "too large" message.
- Decimals like 5.9 and exponent forms like 1e2 are rejected with a message about whole numbers.
- A blank box and a negative number each get their own message. The tool never uses pop-ups or silently changes your input.
- The tool counts groups of distinct objects only.
FAQs About the Combination Calculator
What is the formula for nCr?
The formula is fact(n) divided by fact(r) times fact(n - r). For 12 and 5, that equals 792.
What is the difference between a combination and a permutation?
A combination ignores order. A permutation counts every order separately, so it is larger or equal.
What does "with repetition" mean?
It means you may pick the same item more than once. The tool uses (n + r - 1) choose r for this count, and r may be larger than n.
What happens if r is larger than n?
A red message says the sample size cannot be larger than the total objects. The nCr card shows N/A, but the with-repetition count is still shown. For 3 flavors and 5 scoops, it is 21.
How many five-card poker hands are there?
Enter n = 52 and r = 5. The tool returns 2,598,960 for nCr.
Can the tool handle very large numbers?
Yes, up to 100. It uses big integers, so 100 choose 50 shows every digit with no rounding.
This calculator gives general estimates only and is not professional advice. Reviewed for general accuracy by Prof. Dr. Khalil Mudassar, PhD.