Why is P(n,r) always ≥ C(n,r)?

Because P(n,r) = C(n,r) × r!. Each combination of r items can be arranged in r! different orders. So P(n,r) counts each combination multiple times (once per arrangement).

P(n,n) = n! - the number of ways to arrange all n items. This is the total number of permutations. For example, 5 people can sit in 5 chairs in 5! = 120 ways.

What is the stars and bars method?

Stars and bars is a technique where r identical items (stars) are separated by n−1 dividers (bars) into n groups. The number of arrangements is C(n+r−1, r). Example: ★★|★|★★ represents (2,1,2) distribution.

Why does repetition make the count larger?

Without repetition, each choice reduces the pool. With repetition, you always have n options at each step. P(10,4) = 5,040 without repetition, but 10^4 = 10,000 with repetition.

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Combination & Permutation Calculator

Calculate nCr, nPr, with and without repetition. Step-by-step factorial breakdown and Python code.

Enter Values

n - Total items

r - Items to choose

Sample Data:

Enter n and r, then click Calculate.

Quick Reference:

C(n,r) = n! / (r!(n−r)!) - order doesn't matter
P(n,r) = n! / (n−r)! - order matters
n^r - with repetition, order matters

Frequently Asked Questions

What is the difference between combination and permutation?

Combinations ignore order (choosing team members), permutations care about order (assigning ranks). C(5,3) = 10 but P(5,3) = 60 because each combination of 3 items has 3! = 6 arrangements.

How do I know when to use combinations?

Use combinations when the order of selection doesn't matter. Keywords: "choose", "select", "committee", "subset", "group". Use permutations when order matters: "arrange", "rank", "sequence", "password".

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