Combination Calculator – Guide & Formulas
Calculate combinations (nCr) instantly with our free online combination calculator. Find how many ways to choose items where order does not matter. No signup required!
Try the free calculator
Put these formulas into practice with our instant, step-by-step Combination Calculator.
Free online **combination calculator** — **calculate combinations (nCr)** instantly with step-by-step results. Our **nCr calculator** helps you find how many ways to choose items from a group where **order does not matter**. Whether you need to **calculate combinations** for lottery odds, team selections, card games, or probability problems, this tool provides accurate results with detailed explanations.
Key Takeaway
Use the free Combination Calculator to calculate combinations (ncr) instantly with our free online combination calculator. find how many ways to choose items where order does not matter. no signup required!. Get instant results with step-by-step explanations.
How to Use the Combination Calculator
- Enter the total number of items (n) in the first field.
- Enter the number of items to choose (r) in the second field.
- Click "Calculate Combinations" to see the **nCr formula** and result.
- Review the step-by-step **combination calculation** breakdown.
- Copy or share your combination results.
The Formula
Variable Definitions
- n: Total number of items in the population (the group size)
- r: Number of items to choose or select from the group
- nCr: The number of unique combinations — also written as C(n,r) or "n choose r"
- n!: n factorial — the product of all positive integers up to n
Selecting a 3-Person Team from 8 Employees
Calculate how many ways to choose a team of 3 from 8 candidates using the combination calculator.
- Step 1: Set n = 8 (total employees) and r = 3 (team size).
- Step 2: Apply the combination formula: C(8,3) = 8! / (3! × 5!).
- Step 3: Calculate 8! / 5! = 8 × 7 × 6 = 336.
- Step 4: Divide by 3! = 6: 336 / 6 = 56.
- Step 5: There are 56 unique combinations — 56 different ways to form the team.
Frequently Asked Questions
What is a combination?
A combination is a selection of items from a larger set where the order of selection does not matter. For example, choosing 3 books from a shelf of 5 books is a combination because picking books A, B, C is the same as picking C, A, B. The combination formula calculates how many different ways you can make such selections. The calculator computes combinations for any values of n (total items) and k (items chosen).
What is the combination formula?
The combination formula is C(n,k) = n! / (k!(n-k)!), where n! means n factorial (n times (n-1) times (n-2) times ... times 1). This formula counts the number of ways to choose k items from n items without regard to order. For example, C(5,3) = 5! / (3!2!) = 120 / (6 times 2) = 10. The calculator implements this formula with step-by-step computation.
What does nCr mean?
nCr stands for "n choose r" and is notation for combinations. It represents the number of ways to choose r items from n items without regard to order. For example, 5C3 = C(5,3) = 10. The calculator accepts nCr notation and computes the result using the combination formula.
What is the difference between combinations and permutations?
Combinations count selections where order does not matter, while permutations count arrangements where order matters. Choosing 3 books from 5 is a combination (10 ways). Arranging 3 books from 5 is a permutation (60 ways). Permutations are always larger than combinations for the same n and k. The calculator can compute both.
How does the combination calculator work?
Enter the total number of items (n) and the number to choose (k), and the calculator computes C(n,k). It displays the formula with substituted values, the factorial calculations, and the final result. For large numbers, it uses optimized algorithms to avoid overflow. The calculator also shows the result in different forms (decimal, fraction, scientific notation).
What is C(n,0)?
C(n,0) = 1 for any positive integer n. This makes sense because there is exactly one way to choose zero items from a set: choose nothing. Mathematically, n! / (0!n!) = n! / (1 times n!) = 1. The calculator confirms this property.
What is C(n,n)?
C(n,n) = 1 for any positive integer n. There is exactly one way to choose all n items from a set: take everything. Mathematically, n! / (n!0!) = n! / (n! times 1) = 1. The calculator confirms this property.
What is C(n,1)?
C(n,1) = n for any positive integer n. There are n ways to choose one item from a set of n items. Mathematically, n! / (1!(n-1)!) = n times (n-1)! / (1 times (n-1)!) = n. The calculator confirms this property.
What is C(n,n-1)?
C(n,n-1) = n for any positive integer n. Choosing n-1 items is equivalent to choosing which 1 item to leave out, so there are n choices. Mathematically, n! / ((n-1)!1!) = n. The calculator confirms this property.
What is C(n,k) when k > n?
C(n,k) = 0 when k > n because you cannot choose more items than are available. For example, C(3,5) = 0 because you cannot choose 5 items from a set of 3. The calculator returns 0 for this case.
How do I calculate C(10,3)?
C(10,3) = 10! / (3!7!) = (10 times 9 times 8) / (3 times 2 times 1) = 720 / 6 = 120. There are 120 ways to choose 3 items from 10. The calculator shows the step-by-step computation.
How do I calculate C(52,5)?
C(52,5) = 52! / (5!47!) = (52 times 51 times 50 times 49 times 48) / (5 times 4 times 3 times 2 times 1) = 311875200 / 120 = 2598960. This is the number of possible 5-card poker hands from a standard 52-card deck.
What is C(100,2)?
C(100,2) = 100! / (2!98!) = (100 times 99) / (2 times 1) = 9900 / 2 = 4950. There are 4950 ways to choose 2 items from 100. This is commonly used in probability problems.
What is the symmetry property of combinations?
C(n,k) = C(n,n-k). This means choosing k items is equivalent to choosing which n-k items to leave out. For example, C(10,3) = C(10,7) = 120. The calculator demonstrates this property.
What is the recursive formula for combinations?
C(n,k) = C(n-1,k-1) + C(n-1,k). This recursive relationship is the basis for Pascal's triangle. For example, C(5,2) = C(4,1) + C(4,2) = 4 + 6 = 10. The calculator can show both direct and recursive computation.
What is Pascal's triangle?
Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The nth row contains the values C(n,0), C(n,1), ..., C(n,n). The calculator can generate rows of Pascal's triangle.
How do combinations relate to the binomial theorem?
The binomial theorem states (a+b)^n = sum of C(n,k) times a^(n-k) times b^k for k from 0 to n. The coefficients are the combination values. For example, (a+b)^3 = C(3,0)a^3 + C(3,1)a^2b + C(3,2)ab^2 + C(3,3)b^3 = a^3 + 3a^2b + 3ab^2 + b^3.
What is the sum of all combinations C(n,k) for k from 0 to n?
The sum equals 2^n. This is because each of the n items can either be included or excluded from a subset, giving 2^n total subsets. For example, sum of C(3,k) for k from 0 to 3 = 1 + 3 + 3 + 1 = 8 = 2^3. The calculator verifies this property.
What is C(n,k) with repetition allowed?
When repetition is allowed, the formula changes to C(n+k-1,k) = (n+k-1)! / (k!(n-1)!). This counts the number of ways to choose k items from n types where you can choose the same item multiple times. The calculator supports both with and without repetition.
How do I use the calculator for lottery odds?
For a lottery where you choose k numbers from n total numbers, the odds of winning are 1/C(n,k). For example, choosing 6 numbers from 49 gives C(49,6) = 13983816, so the odds are 1 in 13,983,816. The calculator computes these odds.
What is C(20,10)?
C(20,10) = 20! / (10!10!) = 184756. This is the number of ways to choose 10 items from 20. The calculator shows the step-by-step computation.
What is C(15,5)?
C(15,5) = 15! / (5!10!) = 3003. This is commonly used in probability problems involving selecting 5 items from 15.
What is C(8,4)?
C(8,4) = 8! / (4!4!) = 70. This is the number of ways to choose 4 items from 8.
What is C(12,3)?
C(12,3) = 12! / (3!9!) = 220. This is the number of ways to choose 3 items from 12.
What is C(7,2)?
C(7,2) = 7! / (2!5!) = 21. This is the number of ways to choose 2 items from 7.
How do combinations apply to poker hands?
Poker hand counts use combinations. Total 5-card hands from 52 cards: C(52,5) = 2598960. Four of a kind: C(13,1) times C(4,4) times C(12,1) times C(4,1) = 624. Full house: C(13,1) times C(4,3) times C(12,1) times C(4,2) = 3744. The calculator computes hand probabilities.
How do combinations apply to coin tosses?
In n coin tosses, the number of ways to get exactly k heads is C(n,k). For 10 tosses with exactly 6 heads: C(10,6) = 210. Total outcomes: 2^10 = 1024. Probability of exactly 6 heads: 210/1024 = 20.51%. The calculator computes these probabilities.
What is the relationship between combinations and binomial coefficients?
The binomial coefficient (n choose k) is exactly the combination C(n,k). They are the same mathematical concept. The notation varies by field: mathematics uses C(n,k), statistics uses binomial coefficients, and computer science uses nCk. The calculator uses all notations.
What is C(n,k) when n or k is zero?
C(0,0) = 1 (one way to choose nothing from nothing). C(n,0) = 1 for any n (one way to choose nothing). C(0,k) = 0 for k > 0 (cannot choose items from an empty set). The calculator handles these edge cases.
How do combinations apply to committee selection?
If you need to choose a committee of k people from a group of n people, the number of possible committees is C(n,k). For example, choosing a 3-person committee from 10 people: C(10,3) = 120 possible committees. The calculator computes committee selection probabilities.
What is C(6,3)?
C(6,3) = 6! / (3!3!) = 20. This is the number of ways to choose 3 items from 6.
What is C(9,2)?
C(9,2) = 9! / (2!7!) = 36. This is the number of ways to choose 2 items from 9.
What is C(14,4)?
C(14,4) = 14! / (4!10!) = 1001. This is the number of ways to choose 4 items from 14.
What is C(11,6)?
C(11,6) = 11! / (6!5!) = 462. Using symmetry: C(11,6) = C(11,5) = 462.
What is C(16,8)?
C(16,8) = 16! / (8!8!) = 12870. This is a large combination value.
How do combinations apply to passwords?
If a password uses k distinct characters chosen from n possibilities (order doesn't matter for selection, but does for arrangement), combinations count the selection step. For a 4-character password from 26 letters: C(26,4) = 14950 ways to choose which 4 letters, then 4! = 24 arrangements. Total: 14950 times 24 = 358800.
What is C(18,9)?
C(18,9) = 18! / (9!9!) = 48620. This is the number of ways to choose 9 items from 18.
How do combinations apply to team sports?
If a coach needs to choose k players from n players for a team, the number of possible teams is C(n,k). For choosing 5 starters from 12 players: C(12,5) = 792 possible starting lineups. The calculator computes team selection probabilities.
What is C(13,5)?
C(13,5) = 13! / (5!8!) = 1287. This is the number of ways to choose 5 items from 13.
What is C(20,6)?
C(20,6) = 20! / (6!14!) = 38760. This is the number of ways to choose 6 items from 20.
How do combinations apply to bridge hands?
A bridge hand is 13 cards from 52. The number of possible hands is C(52,13) = 635013559600 (about 635 billion). The probability of a specific hand type uses combinations of suits and ranks. The calculator computes bridge hand probabilities.
What is C(25,12)?
C(25,12) = 25! / (12!13!) = 5200300. This is the number of ways to choose 12 items from 25.
What is C(30,15)?
C(30,15) = 30! / (15!15!) = 155117520. This is a very large combination value.
How do combinations apply to license plates?
If a license plate has k positions each chosen from n options (with repetition), the count is n^k. But if positions must be distinct (no repetition), it involves permutations. Combinations apply when order doesn't matter, such as choosing which k digits to include.
What is C(40,20)?
C(40,20) = 40! / (20!20!) = 137846528820. This is an extremely large combination value, approximately 137.8 billion.
How do combinations apply to genetics?
In genetics, combinations count possible allele combinations. For k gene variants chosen from n possibilities, C(n,k) counts the combinations. This helps calculate genetic diversity and inheritance probabilities. The calculator applies to genetic probability problems.
What is C(50,25)?
C(50,25) = 50! / (25!25!) = 126410606437752. This is approximately 126.4 trillion, one of the largest commonly computed combinations.
How do combinations apply to scheduling?
If you need to choose k time slots from n available slots (order doesn't matter), the count is C(n,k). For choosing 3 meeting times from 10 available slots: C(10,3) = 120 scheduling options. The calculator helps with scheduling problems.
What is C(100,50)?
C(100,50) = 100! / (50!50!) = 100891344545564193334812497256. This is an astronomically large number, approximately 1.009 times 10^29.
How do combinations apply to card games?
Card game probabilities heavily use combinations. In blackjack, the number of 2-card hands is C(52,2) = 1326. In bridge, C(52,13) gives possible hands. In poker, C(52,5) gives possible 5-card hands. The calculator computes card game probabilities.
What is C(35,7)?
C(35,7) = 35! / (7!28!) = 6724520. This is the number of ways to choose 7 items from 35.
What is C(45,9)?
C(45,9) = 45! / (9!36!) = 886163135. This is the number of ways to choose 9 items from 45.
How do combinations apply to computer science?
Combinations appear in algorithm analysis, data structures, and combinatorics. For example, the number of subsets of a set of size n is 2^n (sum of all C(n,k)). Combinations also appear in hash table analysis, network design, and error-correcting codes.
What is C(60,3)?
C(60,3) = 60! / (3!57!) = 34220. This is the number of ways to choose 3 items from 60.
What is C(80,4)?
C(80,4) = 80! / (4!76!) = 1581580. This is the number of ways to choose 4 items from 80.
How do combinations apply to probability?
Probability often uses combinations to count favorable outcomes. For example, probability of drawing 2 aces from a deck: C(4,2)/C(52,2) = 6/1326 = 1/221. The calculator computes probability ratios using combinations.
What is C(70,10)?
C(70,10) = 70! / (10!60!) = 396704524216. This is the number of ways to choose 10 items from 70.
How do combinations apply to survey sampling?
In survey sampling, combinations count possible samples. If selecting k respondents from n people, C(n,k) gives the number of possible samples. Simple random sampling uses combinations to ensure every sample has equal probability.
What is C(24,12)?
C(24,12) = 24! / (12!12!) = 2704156. This is the number of ways to choose 12 items from 24.
What is C(16,6)?
C(16,6) = 16! / (6!10!) = 8008. This is the number of ways to choose 6 items from 16.
How do combinations apply to network design?
In network design, combinations count possible connections. If connecting k nodes from n nodes, C(n,k) gives the number of possible connections. This helps analyze network redundancy and reliability.
What is C(28,14)?
C(28,14) = 28! / (14!14!) = 40116600. This is the number of ways to choose 14 items from 28.
What is C(22,11)?
C(22,11) = 22! / (11!11!) = 705432. This is the number of ways to choose 11 items from 22.
How do combinations apply to quality control?
In quality control, combinations count inspection samples. If inspecting k items from a batch of n, C(n,k) gives possible samples. This helps design sampling plans for quality assurance.
What is C(32,16)?
C(32,16) = 32! / (16!16!) = 601080390. This is the number of ways to choose 16 items from 32.
What is C(18,6)?
C(18,6) = 18! / (6!12!) = 18564. This is the number of ways to choose 6 items from 18.
How do combinations apply to chemistry?
In chemistry, combinations count molecular configurations. For k atoms chosen from n types, C(n,k) counts possible molecules. This helps predict chemical diversity and reaction products.
What is C(36,18)?
C(36,18) = 36! / (18!18!) = 9075135300. This is approximately 9.075 billion, a very large combination value.
What is C(42,21)?
C(42,21) = 42! / (21!21!) = 538257874440. This is approximately 538.3 billion.
How do combinations apply to finance?
In finance, combinations model portfolio selection. Choosing k assets from n possibilities: C(n,k) gives the number of possible portfolios. This helps analyze diversification strategies and risk management.
What is C(26,13)?
C(26,13) = 26! / (13!13!) = 10400600. This is the number of ways to choose 13 items from 26.
What is C(34,17)?
C(34,17) = 34! / (17!17!) = 2333606220. This is approximately 2.33 billion.
How do combinations apply to biology?
In biology, combinations count species groupings. For k species chosen from n in a study, C(n,k) counts possible groupings. This helps in biodiversity analysis and ecological surveys.
What is C(38,19)?
C(38,19) = 38! / (19!19!) = 35345263800. This is approximately 35.3 billion.
What is C(44,22)?
C(44,22) = 44! / (22!22!) = 210905691636. This is approximately 210.9 billion.
How do combinations apply to education?
In education, combinations count course selections. Choosing k courses from n available: C(n,k) gives possible schedules. This helps students plan course loads and administrators predict enrollment patterns.
What is C(48,24)?
C(48,24) = 48! / (24!24!) = 32247603683100. This is approximately 32.2 trillion.
What is C(54,27)?
C(54,27) = 54! / (27!27!) = 291466956972612. This is approximately 291.5 trillion.
How do combinations apply to music?
In music, combinations count note selections. Choosing k notes from n available: C(n,k) gives possible chord combinations. This helps in music theory analysis and composition.
What is C(56,28)?
C(56,28) = 56! / (28!28!) = 1541430808034820. This is approximately 1.54 quadrillion.
What is C(58,29)?
C(58,29) = 58! / (29!29!) = 30067266499541040. This is approximately 30.07 quadrillion.
How do combinations apply to architecture?
In architecture, combinations count design options. Choosing k materials from n available: C(n,k) gives possible combinations. This helps architects explore design possibilities and material selection.
What is C(62,31)?
C(62,31) = 62! / (31!31!) = 465428353255261088. This is approximately 465.4 quadrillion.
What is C(64,32)?
C(64,32) = 64! / (32!32!) = 1832624140942590534. This is approximately 1.83 quintillion.
How do combinations apply to game theory?
In game theory, combinations count strategy profiles. For k strategies chosen from n: C(n,k) gives possible strategy combinations. This helps analyze game equilibria and player strategies.
What is C(66,33)?
C(66,33) = 66! / (33!33!) = 7219428434016265740. This is approximately 7.22 quintillion.
What is C(68,34)?
C(68,34) = 68! / (34!34!) = 28328631172127521020. This is approximately 28.33 quintillion.
How do combinations apply to linguistics?
In linguistics, combinations count word pairings. For k words chosen from n: C(n,k) counts possible pairings. This helps analyze language patterns and phonetic combinations.
What is C(72,36)?
C(72,36) = 72! / (36!36!) = 425188768206673931720. This is approximately 425.2 quintillion.
What is C(74,37)?
C(74,37) = 74! / (37!37!) = 1631878930337487168920. This is approximately 1.63 quintillion.
How do combinations apply to social sciences?
In social sciences, combinations count group formations. For k people chosen from n: C(n,k) counts possible groups. This helps analyze social networks and community structures.
What is C(76,38)?
C(76,38) = 76! / (38!38!) = 6242595062835966189680. This is approximately 6.24 quintillion.
What is C(78,39)?
C(78,39) = 78! / (39!39!) = 23904945648280749131640. This is approximately 23.9 quintillion.
How do combinations apply to environmental science?
In environmental science, combinations count species combinations. For k species chosen from n in an ecosystem: C(n,k) counts possible assemblages. This helps predict biodiversity and ecosystem resilience.
What is C(82,41)?
C(82,41) = 82! / (41!41!) = 35410753628955309982900. This is approximately 35.4 quintillion.
What is C(84,42)?
C(84,42) = 84! / (42!42!) = 135177959818389094178840. This is approximately 135.2 quintillion.
How do combinations apply to medicine?
In medicine, combinations count treatment options. For k treatments chosen from n: C(n,k) counts possible combinations. This helps design clinical trials and treatment protocols.
What is C(86,43)?
C(86,43) = 86! / (43!43!) = 518263799302765744241160. This is approximately 518.3 quintillion.
What is C(88,44)?
C(88,44) = 88! / (44!44!) = 1988646929697851147782600. This is approximately 1.99 sextillion.
How do I use the combination calculator effectively?
Enter n (total items) and k (items to choose), then click calculate. The calculator shows the formula, step-by-step computation, and final result. Use it for probability problems, lottery odds, poker hands, committee selection, and any problem where order doesn't matter. Remember: combinations (order doesn't matter) vs permutations (order matters).