Calculate permutations and combinations. Find P(n,r) and C(n,r) with step-by-step explanations and examples.
Permutations
720
P(10, 3)
Combinations
120
C(10, 3)
Ways to arrange 3 items from 10, where order matters
720
P(10, 3)
Ways to choose 3 items from 10, where order doesn't matter
120
C(10, 3)
P(n, r) = n! / (n - r)!
P(10, 3) = 10! / (10 - 3)!
P(10, 3) = 10! / 7!
P(10, 3) = 3,628,800 / 5,040
P(10, 3) = 720
C(n, r) = n! / (r! × (n - r)!)
C(10, 3) = 10! / (3! × (10 - 3)!)
C(10, 3) = 10! / (3! × 7!)
C(10, 3) = 3,628,800 / (6 × 5,040)
C(10, 3) = 120
Key insight: P(n,r) = C(n,r) × r! because each combination can be arranged r! ways to form permutations.
720 = 120 × 6
Permutations
720
P(10, 3)
Combinations
120
C(10, 3)
Permutations count ordered arrangements where order matters: nPr = n!/(n-r)!. Combinations count selections where order does not matter: nCr = n!/((n-r)! x r!). For example, choosing 3 from 5: permutations = 60 (order matters), combinations = 10 (order does not matter). Use permutations for arrangements, combinations for selections.
Permutations and combinations are counting techniques in combinatorics. Permutations count the number of ways to arrange items where order matters (like passwords or race positions). Combinations count the number of ways to select items where order does not matter (like lottery numbers or committee members). Both use factorial notation.
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Test your understanding of permutations
Practice with 4 problems to test your understanding.
Permutations count ordered arrangements where order matters: nPr = n!/(n-r)!. Combinations count selections where order does not matter: nCr = n!/((n-r)! x r!). For example, choosing 3 from 5: permutations = 60 (order matters), combinations = 10 (order does not matter). Use permutations for arrangements, combinations for selections.
A permutation is an arrangement of objects where ORDER MATTERS. P(n,r) counts ways to arrange r items from n total items. Example: Arranging 3 books on a shelf from 5 books = P(5,3) = 60 different arrangements.
A combination is a selection of objects where ORDER DOESN'T MATTER. C(n,r) counts ways to choose r items from n items. Example: Choosing 3 people from 5 for a committee = C(5,3) = 10 different groups.
Permutation: ORDER MATTERS (ABC ≠ BAC). Use for rankings, arrangements, sequences. Combination: ORDER DOESN'T MATTER (ABC = BAC). Use for selections, committees, groups. P(n,r) is always ≥ C(n,r).
Factorial (n!) is the product of all positive integers up to n. 5! = 5×4×3×2×1 = 120. It counts all possible arrangements of n distinct objects. By definition, 0! = 1.
Use permutations when: arranging items in order, assigning positions, creating passwords. Use combinations when: forming committees, choosing lottery numbers, selecting items from a menu.
Last updated: 2025-01-15
Permutations
720
P(10, 3)
Combinations
120
C(10, 3)