حاسبة التباديل والتوافيق
The detailed guide below is currently available in English.
Permutations versus combinations
Both formulas count ways to choose r items from n, and the entire difference is whether order is recorded. Permutations count arrangements: choosing 3 of 10 runners for gold-silver-bronze gives 10P3 = 10 × 9 × 8 = 720, because each of the three slots shrinks the pool. Combinations count groups where order is ignored, so the same 10 choose 3 gives 10C3 = 720 ÷ 3! = 120 — each 3-person group was counted 3! = 6 times as an arrangement. That division by r! is the whole bridge between the two, and it is why combinations never exceed permutations.
الأسئلة الشائعة
How do I decide whether to use nPr or nCr?
Ask if swapping two chosen items creates a new outcome. Seats, ranks, and password characters are permutations; committees, hands of cards, and pizza toppings are combinations.
Why does choosing all n items give 1 combination but n! permutations?
Taking everything leaves exactly one group — nCn = 1 — but that group can be arranged in n! different orders, which is precisely how the formula divides n! by itself.