N Choose K Calculator – Combinations and Permutations
Calculate combinations and permutations (n choose k).
Table of Contents
How to Use
- Enter n (the total number of items)
- Enter k (the number of items to choose)
- Click calculate to find combinations and permutations
- View the formulas and results
What is N Choose K?
N choose K, written as C(n,k) or (n k), is the binomial coefficient that represents the number of ways to choose k items from n items without regard to order. It's also called 'n choose k' or 'combinations'.
The formula is: C(n,k) = n! / (k! × (n-k)!)
Combinations vs Permutations
The key difference is whether order matters:
- Combinations: Order doesn't matter. Choosing {A, B, C} is the same as {C, B, A}
- Permutations: Order matters. ABC is different from CBA
- Permutations are always greater than or equal to combinations
- P(n,k) = C(n,k) × k!
Real-World Examples
Combinations and permutations appear in many situations:
- Lottery: How many ways to pick 6 numbers from 49? C(49,6) = 13,983,816
- Poker hands: 5 cards from 52 = C(52,5) = 2,598,960 possible hands
- Team selection: Choosing 5 players from 12 = C(12,5) = 792 ways
- Password arrangements: Arranging 4 digits = P(10,4) = 5,040 permutations
Properties of Binomial Coefficients
- C(n,0) = C(n,n) = 1
- C(n,k) = C(n, n-k) (symmetry)
- C(n,k) = C(n-1,k-1) + C(n-1,k) (Pascal's identity)
- Sum of row n in Pascal's triangle = 2^n
Frequently Asked Questions
- When should I use combinations vs permutations?
- Use combinations when the order of selection doesn't matter (like choosing team members). Use permutations when order matters (like arranging people in a line or assigning positions).
- Why is C(n,k) = C(n, n-k)?
- Choosing k items to include is the same as choosing (n-k) items to exclude. For example, choosing 3 people from 5 to be on a team is equivalent to choosing 2 people to not be on the team.
- What is Pascal's Triangle?
- Pascal's Triangle is a triangular array where each number is the sum of the two numbers above it. The nth row contains all binomial coefficients C(n,0) through C(n,n).
- Can n choose k handle large numbers?
- This calculator uses arbitrary precision arithmetic to handle very large numbers accurately. However, extremely large values may take longer to compute.