Calculate P(n,k) and C(n,k) with Step-by-Step Derivation
Total Items n
Selected Items k
Result
Step-by-Step Derivation
Permutation & Combination Formulas
Permutation: P(n,k) = n! / (n-k)!
Combination: C(n,k) = n! / (k! × (n-k)!)
Factorial: n! = n × (n-1) × ... × 2 × 1
Permutation considers order, combination does not. Both are fundamental in probability, statistics and discrete math.
⚠n and k must be non-negative integers, k ≤ n. Large values will show in scientific notation.
What are Permutation & Combination?
Permutation: select k items from n and arrange in order (order matters). Combination: select k items without considering order (order does not matter). Widely used in probability, algorithms and statistics.
Permutation
P(n,k)=n!/(n-k)! Example: choose president & vice-president from 3 students: P(3,2)=6.
Combination
C(n,k)=n!/(k!(n-k)!) Example: choose 2 representatives from 3 students: C(3,2)=3.
Factorial
n! = total permutations of n items. 0! = 1 (defined).
Binomial Coefficient
C(n,k) = coefficient in binomial expansion (a+b)ⁿ. Can be calculated with Pascal’s triangle.
What is the difference between permutation and combination?▼
Permutation considers order (AB ≠ BA), combination does not (AB = BA). Formula: P(n,k)=n!/(n-k)! , C(n,k)=n!/(k!(n-k)!). Choosing president & vice-president is permutation; choosing committee members is combination.
Are P(n,k) and A(n,k) the same?▼
Yes, both represent permutations. A(n,k) (Arrangement) is common in Chinese textbooks, P(n,k) (Permutation) is international standard. Formula: P(n,k)=A(n,k)=n!/(n-k)!.
What is 0! and why?▼
0! = 1 (mathematical definition). There is exactly 1 way to arrange an empty set. The formula C(n,0)=1 requires 0! = 1 to be valid.
Real-life applications of permutation & combination?▼
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