What is the difference between permutation and combination?

Q: What is the difference between permutations and combinations?

Permutations consider order (ABC ≠ ACB) while combinations don't (ABC = ACB). For choosing 3 from 5: permutations = 5!/(5-3)! = 60, combinations = 5!/(3!(5-3)!) = 10. Use permutations when arrangement matters (passwords, seating), combinations when it doesn't (team selection).

Q: What is the formula for permutations?

The number of permutations of n items taken r at a time is P(n,r) = n!/(n-r)!. For example, P(5,3) = 5!/(5-3)! = 120/2 = 60. This means there are 60 ways to arrange 3 items selected from 5. If repetitions are allowed, it's n^r = 5^3 = 125.

Q: When would I use permutations in real life?

Permutations are used for: calculating password combinations (though technically permutations), scheduling/ordering tasks, determining race placements, arranging books on a shelf, creating seating arrangements, phone number possibilities, and any scenario where the order of selection matters.

Q: What does 'with repetition' mean in permutations?

Permutations with repetition allow items to be selected more than once. For example, a 4-digit PIN has 10^4 = 10,000 possible permutations (0000-9999) because digits can repeat. Without repetition (like arranging 4 unique books), it's P(10,4) = 5,040. The calculator supports both modes.

Q: How do factorials work in permutation calculations?

A factorial (n!) is the product of all positive integers up to n: 5! = 5×4×3×2×1 = 120. Factorials grow very fast: 10! = 3,628,800 and 20! = 2.4×10^18. This is why the number of possible arrangements becomes astronomically large even for small sets of items.

Question

What is the difference between permutation and combination?

Accepted Answer

Permutation (nPr): Order matters. n!/(n-r)!. Combination (nCr): Order does not matter. n!/[r!(n-r)!]. For example, ABC and ACB are different permutations but the same combination.

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