What N Choose K Means
N choose k is the number of ways to choose k items from n total items when the order does not matter. It is one of the most common formulas in probability, statistics, combinatorics, and card-counting problems.
Combination vs Permutation
The important difference is order. In combinations, choosing A, B, and C is the same as choosing C, B, and A. In permutations, those different orders are counted separately.
| Type | Order matters? | Example |
|---|---|---|
| Combination | No | Selecting 3 students from a class. |
| Permutation | Yes | Ranking 3 students in 1st, 2nd, and 3rd place. |
Why This Calculator Uses a Multiplicative Method
The direct factorial formula is simple, but factorials become very large very quickly. For example, 100! is already enormous. Instead of calculating huge factorials separately, this calculator uses a safer exact integer method.
The calculator also uses the symmetry rule:
This means C(100,97) is calculated as C(100,3), which is much faster and easier.
Example Calculation
Choose 3 items from 5: C(5,3) = 5! / [3! × (5−3)!] C(5,3) = 5! / (3! × 2!) C(5,3) = 120 / (6 × 2) C(5,3) = 10
Common Uses
- Lottery probability calculations.
- Card game hand counting.
- Choosing teams or committees.
- Binomial probability problems.
- Statistics and combinatorics homework.
- Counting possible subsets of a larger group.
Common Mistakes
- Using combinations when order actually matters.
- Entering k greater than n.
- Trying to calculate large factorials directly with normal floating-point numbers.
- Forgetting that C(n,0) and C(n,n) both equal 1.
- Confusing C(n,k) with n × k.
Frequently Asked Questions
What is n choose k?
It is the number of ways to choose k items from n items without considering order.
What is C(n,0)?
C(n,0) = 1. There is exactly one way to choose nothing from a set.
What is C(n,n)?
C(n,n) = 1. There is exactly one way to choose all items from the set.
Can this calculator handle large values?
Yes, it uses BigInt for exact integer arithmetic. Very large values may still produce extremely long outputs, but the calculation avoids normal JavaScript floating-point precision loss.
What is the difference between combinations and permutations?
Combinations ignore order. Permutations count different orders separately.