Combinations Calculator

Combinations and permutations count the number of ways to select or arrange items from a group. The critical distinction is whether order matters. Choosing a committee of 3 from 10 people: order does not matter, so use combinations. Arranging 3 books in order on a shelf from a set of 10: order matters, so use permutations. Combinations (nCr) are also called binomial coefficients and appear in Pascal's triangle, probability theory, and the binomial theorem. Permutations (nPr) count ordered arrangements and are used in password counting, race outcomes, and scheduling. Both use factorial notation. n! means the product of all integers from 1 to n. The calculator computes these iteratively to handle n up to 170 before JavaScript floating-point overflow.

• Lottery odds: A lottery requires picking 6 numbers from 49. The number of possible tickets is C(49,6) = 13,983,816. Probability of winning = 1 / 13,983,816.
• Committee selection: How many ways can a 5-person committee be chosen from 20 candidates? C(20,5) = 15,504 possible committees.
• Password counting: A 4-digit PIN from digits 0-9 without repetition has P(10,4) = 5,040 possible combinations (order matters, no repeats).
• Tournament brackets: How many ways can 3 medals be awarded to 8 competitors? P(8,3) = 336 possible gold-silver-bronze outcomes.

1. 1Enter n (the total number of items to choose from) and r (the number being chosen or arranged).
2. 2The calculator validates that both are non-negative integers and that r does not exceed n.
3. 3It computes n!, r!, and (n−r)! iteratively to avoid recursion overhead.
4. 4nCr = n! / (r! × (n−r)!) counts unordered selections.
5. 5nPr = n! / (n−r)! counts ordered arrangements.
6. 6All factorial values are shown in the secondary results.

• Confusing order matters vs. order does not matter: The most common error. Picking a winning lottery ticket (unordered) uses combinations. Placing 1st through 3rd in a race (ordered) uses permutations. Ask: does swapping the selection change the outcome?
• Entering non-integer values: Factorials are only defined for non-negative integers. Entering n=5.5 or r=2.3 is invalid. The calculator returns an error for non-integer inputs.
• Setting r greater than n: You cannot choose more items than are available. C(5,7) is undefined because r > n. The calculator flags this as an error.
• Overflow for large factorials: 170! is the largest exact factorial JavaScript can represent. For n > 170, the result overflows to Infinity. Use logarithms of factorials for very large combinatorial problems.

• Use C(n,r) = C(n, n−r) to simplify computation: C(100, 97) = C(100, 3) = 161,700. When r is close to n, compute with the smaller of r and n−r to reduce the number of multiplications.
• Verify with Pascal's triangle for small values: Pascal's triangle gives C(n,r) directly. Row n, position r (starting at 0). Row 5: 1, 5, 10, 10, 5, 1. So C(5,2) = 10. Use this to spot-check the calculator for small inputs.
• nPr = nCr × r!: Permutations equal combinations times r factorial. Every combination can be arranged in r! ways. This relationship is worth memorizing as a quick sanity check.
• For very large n, use Stirling's approximation: When n exceeds 170, switch to logarithmic methods: ln(n!) ≈ n×ln(n) − n + 0.5×ln(2πn). This gives log-probabilities without overflow.