Calculate C N K Pk 1-P N-K

This calculator helps you compute combinations (C(n,k)), permutations (Pk), and the probability of at least k successes in n trials (1-P(n-k)). These calculations are fundamental in probability theory and combinatorics.

What is C(n,k)?

The combination formula C(n,k) calculates the number of ways to choose k items from a set of n items without regard to order. It's also known as "n choose k."

Formula: C(n,k) = n! / (k! × (n-k)!)

Where:

n! = factorial of n (n × (n-1) × ... × 1)
k! = factorial of k
(n-k)! = factorial of (n-k)

Combinations are used in probability, statistics, and combinatorial mathematics to determine the number of possible outcomes in scenarios where order doesn't matter.

What is Pk?

Pk represents the number of permutations of k items from a set of n items, where order matters. Unlike combinations, permutations count different arrangements as distinct.

Formula: Pk = n! / (n-k)!

Permutations are used in probability, cryptography, and scheduling problems where the sequence of items matters.

What is 1-P(n-k)?

1-P(n-k) calculates the probability of getting at least k successes in n independent Bernoulli trials, where each trial has a probability p of success.

Formula: 1-P(n-k) = 1 - Σ [C(n,i) × pᵢ × (1-p)^(n-i)] for i=0 to k-1

This is useful in quality control, medical testing, and reliability engineering to determine the probability of achieving a minimum number of successes.

How to Use This Calculator

Enter the total number of items (n) in the first field.
Enter the number of items to choose (k) in the second field.
For the probability calculation, enter the probability of success (p) in the third field.
Click "Calculate" to see the results for C(n,k), Pk, and 1-P(n-k).
Review the results and interpretation provided.

Note: All inputs must be positive integers for C(n,k) and Pk. For 1-P(n-k), p must be between 0 and 1.

Examples

Example 1: Combinations

If you have 5 cards and want to know how many ways you can choose 2 cards (order doesn't matter), the calculation is C(5,2) = 10.

Example 2: Permutations

If you have 4 books and want to arrange 2 of them on a shelf (order matters), the calculation is Pk = 12.

Example 3: Probability

If you flip a fair coin 10 times, the probability of getting at least 6 heads is approximately 0.8789.

FAQ

When should I use combinations instead of permutations?

Use combinations when the order of items doesn't matter (e.g., selecting a team from a group). Use permutations when order matters (e.g., arranging books on a shelf).

What is the difference between C(n,k) and Pk?

C(n,k) counts the number of ways to choose k items without regard to order, while Pk counts the number of ways to arrange k items where order matters.

How is 1-P(n-k) different from P(n-k)?

1-P(n-k) represents the probability of getting at least k successes, while P(n-k) represents the probability of getting exactly k successes.

Can I use these calculations for continuous probability distributions?

No, these calculations are designed for discrete probability distributions (e.g., binomial distribution). For continuous distributions, you would use different formulas.