When working with probability, statistics, or combinatorics, you often need to figure out how many different ways you can select a subset of items from a larger group. This is where the N Choose R Calculator (also called a combination calculator) comes into play.
Our free tool makes it simple to calculate nCr = N! / (R! × (N−R)!) in just a few clicks. Whether you’re a student, teacher, researcher, or professional, this calculator will save you time and reduce the chance of errors.
N Choose R Calculator
What is N Choose R (nCr)?
In mathematics, N Choose R represents the number of possible combinations of R items selected from a total of N items without considering the order of selection.
- N = Total number of items
- R = Number of items to choose
- nCr Formula: nCr=N!R!(N−R)!nCr = \frac{N!}{R!(N – R)!}nCr=R!(N−R)!N!
Here, ! (factorial) means multiplying all positive integers up to that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Why Use an N Choose R Calculator?
Manually calculating combinations with factorials can quickly become complex, especially for large values of N. The calculator provides:
- ✅ Instant results – no long factorial expansions.
- ✅ Error-free calculation – reduces mistakes in manual math.
- ✅ Time-saving – perfect for exams, research, and probability problems.
- ✅ Practical applications – in statistics, probability, card games, lotteries, and computer science.
How to Use the N Choose R Calculator
Using our calculator is straightforward:
- Enter N (Total Items) – The total number of items in your dataset.
- Enter R (Items to Choose) – How many items you want to select.
- Click “Calculate” – The tool instantly computes the result.
- View Result – The value of N Choose R will be displayed.
- Click “Reset” – Clear inputs and start a new calculation.
⚠️ Note: You must enter values such that N ≥ R ≥ 0, otherwise the calculator will show an error message.
Example Calculations
Let’s go through a few examples:
Example 1: Choosing a Committee
If you have 10 people and want to select 3 members for a committee: 10C3=10!3!(10−3)!=10×9×83×2×1=12010C3 = \frac{10!}{3!(10-3)!} = \frac{10 × 9 × 8}{3 × 2 × 1} = 12010C3=3!(10−3)!10!=3×2×110×9×8=120
✅ There are 120 different ways to form the committee.
Example 2: Lottery Numbers
If a lottery requires choosing 6 numbers out of 49: 49C6=49!6!(49−6)!=13,983,81649C6 = \frac{49!}{6!(49-6)!} = 13,983,81649C6=6!(49−6)!49!=13,983,816
✅ There are nearly 14 million possible combinations.
Example 3: Simple Selection
If you have 5 fruits and want to pick 2: 5C2=5!2!(5−2)!=5×42=105C2 = \frac{5!}{2!(5-2)!} = \frac{5 × 4}{2} = 105C2=2!(5−2)!5!=25×4=10
✅ You can make 10 unique pairs of fruits.
Applications of N Choose R
The N Choose R formula is widely used across different fields:
- 🎲 Probability theory – calculating event likelihoods.
- 📊 Statistics – sampling without replacement.
- 🧮 Mathematics – solving binomial expansion problems.
- 🃏 Card games – calculating poker hand probabilities.
- 🎟️ Lotteries – determining odds of winning.
- 💻 Computer science – algorithms involving combinations.
Key Benefits of Our Calculator
- Works on all devices (desktop, mobile, tablet).
- Easy input with instant calculation.
- User-friendly design with clear results.
- Helps students and professionals equally.
- Free to use, no sign-up required.
20 Frequently Asked Questions (FAQs)
Q1: What does N Choose R mean?
It means selecting R items from N items without considering order.
Q2: What is the formula for N Choose R?
The formula is nCr=N!/(R!×(N−R)!)nCr = N! / (R! × (N−R)!)nCr=N!/(R!×(N−R)!).
Q3: What does the exclamation mark (!) mean in the formula?
It denotes a factorial, the product of all positive integers up to that number.
Q4: Is N Choose R the same as permutation?
No. Combinations (nCr) ignore order, while permutations (nPr) consider order.
Q5: Can N and R be negative numbers?
No, both must be non-negative integers.
Q6: What if R > N?
That’s not valid. You cannot choose more items than the total available.
Q7: What is 0 Choose 0?
By definition, 0C0=10C0 = 10C0=1.
Q8: How do I calculate 5 Choose 2?
5C2=105C2 = 105C2=10.
Q9: What’s the difference between combinations and arrangements?
Combinations ignore order, while arrangements (permutations) count order.
Q10: Is N Choose R used in probability?
Yes, it is often used in probability and statistics problems.
Q11: What is 10 Choose 3?
10C3=12010C3 = 12010C3=120.
Q12: What is 52 Choose 5 (poker hands)?
52C5=2,598,96052C5 = 2,598,96052C5=2,598,960.
Q13: Why do we divide by R! in the formula?
Because the order of selected items doesn’t matter, so we remove duplicates.
Q14: Can I use this calculator for large values of N and R?
Yes, but extremely large values may result in very big numbers.
Q15: Is N Choose R symmetric?
Yes, nCr=nC(n−r)nCr = nC(n−r)nCr=nC(n−r).
Q16: What is 100 Choose 1?
It equals 100, since choosing 1 from 100 gives 100 possible outcomes.
Q17: Is there a fast way to approximate nCr?
For very large numbers, approximations like Stirling’s formula can be used.
Q18: Where is nCr used in real life?
In lottery systems, card games, research sampling, and probability analysis.
Q19: What’s the difference between 10C3 and 10P3?
10C3 = 120 (combinations), while 10P3 = 720 (permutations).
Q20: Can this calculator handle decimal inputs?
No, it only works with whole numbers (integers).
Final Thoughts
The N Choose R Calculator is a powerful yet simple tool for quickly computing combinations. Instead of manually working through factorials, you can get instant results with just two inputs: N (total items) and R (items to choose).