What is the difference between permutation and combination?+
A permutation counts ordered arrangements - the order of selection matters. A combination counts unordered selections - only which items are chosen matters, not the order. For example, selecting 3 letters from {A, B, C}: permutations give ABC, ACB, BAC, BCA, CAB, CBA (6 results), while combinations give just {A,B,C} (1 result). Choosing a PIN uses permutations; choosing lottery numbers uses combinations.
What is the formula for permutation nPr?+
P(n, r) = n! / (n − r)!, where n is the total number of items and r is how many you select. For example, P(5, 2) = 5! / 3! = 120 / 6 = 20. This counts the number of ways to pick and arrange r items from n distinct items in an ordered sequence.
What is the formula for combination nCr?+
C(n, r) = n! / (r! × (n − r)!). For example, C(5, 2) = 5! / (2! × 3!) = 120 / 12 = 10. Since order is ignored, each group of r items is counted once - exactly r! times fewer than permutations, which is why we divide P(n, r) by r! to get C(n, r).
How do I calculate 10C3 by hand?+
C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120. The shortcut: only compute the top r terms of n!, then divide by r!. The (n-r)! in the denominator cancels with the lower portion of n!. So for any C(n, r), compute (n × (n-1) × ... × (n-r+1)) / r! - far easier than computing full factorials.
What is 0! (zero factorial) and why is it 1?+
0! = 1 by mathematical convention. This ensures C(n, 0) = 1 (there is exactly one way to choose nothing from a set) and C(n, n) = 1 (exactly one way to choose everything). Without this definition, the combinatorics formulas would be undefined at the boundary cases. It also follows naturally from the recursive definition n! = n × (n−1)!: 1! = 1, so 0! = 1!/1 = 1.
When should I use permutations in real life?+
Use permutations when arrangement or sequence matters: awarding 1st/2nd/3rd place among contestants, creating a 4-digit PIN from 0–9 without repetition (P(10,4) = 5,040), scheduling the order of tasks, arranging books on a shelf, anagram counting, and any problem where position, rank, or sequence is meaningful. If swapping two selected items gives a different result, use permutations.
When should I use combinations in real life?+
Use combinations when only the selection group matters: picking lottery numbers (C(49,6) = 13,983,816), forming a committee, choosing pizza toppings, dealing cards, selecting a sample for a study, or choosing which projects to fund from a list. If swapping two selected items gives the same result, use combinations.
What is the symmetry property C(n, r) = C(n, n-r)?+
Choosing r items from n is equivalent to rejecting (n−r) items. C(10, 3) = C(10, 7) = 120. This symmetry is visible in Pascal's triangle, where each row reads the same from left and right. Practically, always compute using the smaller of r and (n−r) to minimize arithmetic: C(100, 97) = C(100, 3) = 161,700, which is far easier to compute directly.
How are combinations used in the binomial theorem?+
The binomial theorem states (a+b)^n = Σ C(n,k) a^(n-k) b^k for k from 0 to n. The coefficients C(n,k) are called binomial coefficients and form Pascal's triangle. For (a+b)^4: coefficients are C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1, giving a⁴+4a³b+6a²b²+4ab³+b⁴. This connection makes combinatorics central to algebra and probability.
What is the maximum n this calculator can handle accurately?+
The calculator uses JavaScript's 64-bit floating point, which represents integers exactly up to 2^53 (about 9 quadrillion). For very large n, factorials overflow to Infinity. In practice, nCr and nPr results remain representable for n up to about 60–70 for exact integer results. The calculator uses a numerically stable approach to delay overflow as long as possible.