Combination Calculator (nCr)
Print PageA Combination Calculator (also known as an nCr Calculator, n Choose r Calculator, Binomial Coefficient Calculator, or Unordered Selection Utility) calculates the total number of unique ways to select a subset of r items from a total set of n distinct items where order does NOT matter (nCr = &leftlpar;nr&rightrpar; = n! ÷ [ r! · (n - r)! ]). Whether you are determining lottery jackpot odds (such as selecting 5 white balls out of 69 in Powerball), evaluating 5-card poker hand probabilities, assembling committee boards from a candidate pool, or calculating Pascal’s Triangle binomial expansions, a combination calculator delivers exact mathematical outputs instantly.
In combinatorics, the fundamental distinction between Combinations and Permutations lies in sequence ordering: for combinations, {A, B, C} is identical to {C, B, A} (order does not matter), whereas for permutations, every distinct ordering counts as a separate outcome (nPr = n! ÷ (n - r)!).
Our free online Combination Calculator provides instant calculations across all four standard selection modes:
- Standard Combinations (Order Does NOT Matter, Without Repetition):
nCr = n! ÷ [ r! · (n - r)! ]. - Standard Permutations (Order DOES Matter, Without Repetition):
nPr = n! ÷ (n - r)! = nCr · r!. - Combinations with Repetition (Stars and Bars Theorem):
nHr = (n + r - 1)! ÷ [ r! · (n - 1)! ]. - Permutations with Repetition:
nr.
Master nCr Combination Lookup Table (For Pool Size n = 10)
The table below displays the exact calculated combinations (nCr), permutations (nPr), and symmetry relationships when selecting r items from a total set of n = 10 distinct items:
| Items Chosen (r) | Combinations nCr (Order Ignored) | Permutations nPr (Order Matters) | Symmetry Property Equation nCr = nC(n-r) | Real-World Practical Example |
|---|---|---|---|---|
| r = 0 Items | 1 way | 1 way | 10C0 = 10C10 = 1 | Choosing an empty set (1 unique option) |
| r = 1 Item | 10 ways | 10 ways | 10C1 = 10C9 = 10 | Selecting 1 representative from 10 members |
| r = 2 Items | 45 ways | 90 ways | 10C2 = 10C8 = 45 | Selecting a 2-person handshake pair |
| r = 3 Items | 120 ways | 720 ways | 10C3 = 10C7 = 120 | Selecting 3 ice cream flavors from 10 |
| r = 4 Items | 210 ways | 5,040 ways | 10C4 = 10C6 = 210 | Selecting a 4-person project committee |
| r = 5 Items (Symmetry Peak) | 252 ways (Peak) | 30,240 ways | 10C5 = 252 | Dividing 10 people into two equal 5-man teams |
| r = 6 Items | 210 ways | 151,200 ways | 10C6 = 10C4 = 210 | Choosing 6 winning lottery numbers from 10 |
| r = 10 Items (All Items) | 1 way | 3,628,800 ways | 10C10 = 10C0 = 1 | Choosing all 10 items (only 1 unique combination) |
Step-by-Step Powerball Lottery & 5-Card Poker Calculation Example
To calculate the exact number of combinations for selecting 5 white balls out of 69 in US Powerball lottery (n = 69, r = 5), and for drawing a 5-card poker hand from a standard 52-card deck (n = 52, r = 5):
Step 1 (Powerball 69C5 Combination Formula): 69C5 = 69! ÷ [ 5! · (69 - 5)! ] = 69! ÷ (5! · 64!)
Step 2 (Cancel Common Factor 64!): 69C5 = (69 × 68 × 67 × 66 × 65) ÷ (5 × 4 × 3 × 2 × 1)
Step 3 (Multiply & Divide): 69C5 = 1,348,621,560 ÷ 120 = 11,238,513 unique 5-ball combinations
Step 4 (Multiply by Red Powerball 26): Total Powerball Jackpot Odds = 11,238,513 × 26 = 292,201,338 (1 in 292.2 Million)
Step 5 (Poker 52C5 Combination Formula): 52C5 = 52! ÷ [ 5! · 47! ] = (52 × 51 × 50 × 49 × 48) ÷ 120 = 311,875,200 ÷ 120 = 2,598,960 unique 5-card hands.
Thus, there are exactly 11,238,513 ways to pick 5 white lottery balls and 2,598,960 possible poker hands.
Combinations (nCr) vs. Permutations (nPr)
Below is a comparative reference chart detailing when to use Combinations versus Permutations based on problem characteristics:
| Selection Metric / Rule | Combinations (nCr) | Permutations (nPr) |
|---|---|---|
| Does Order Matter? | NO (Order is completely ignored) | YES (Order creates distinct outcomes) |
| Exact Mathematical Formula | nCr = n! ÷ [ r! · (n - r)! ] |
nPr = n! ÷ (n - r)! |
| Relationship Between Formulas | nCr = nPr ÷ r! |
nPr = nCr · r! |
| Typical Real-World Applications | Lottery numbers, poker hands, committee selection | Locker PIN codes, race finishing order (1st, 2nd, 3rd) |
History & Mathematics: 6th Century BCE Pingala to 1654 Blaise Pascal
6th Century BCE Ancient India: Sushruta & Pingala
The earliest known systematic calculation of combinations appears in ancient India. In the 6th century BCE medical treatise Sushruta Samhita, physician Sushruta correctly calculated that 6 basic tastes (sweet, sour, salty, pungent, bitter, astringent) can be combined in exactly 26 - 1 = 63 unique ways. In the 2nd century BCE, Indian prosodist Pingala derived combination rules (nCr) in his Chandaś&amacron;stra to count poetic meter variations.
1654 Blaise Pascal & Traité du triangle arithmétique
In 1654, French mathematician Blaise Pascal published Traité du triangle arithmétique, standardizing the properties of Pascal’s Triangle. Pascal proved that the entry in row n and column r equals the binomial coefficient nCr = &leftlpar;nr&rightrpar;.
Popular direct tools:
Frequently Asked Questions (FAQ)
What is the difference between nCr and nPr?
In combinations (nCr), order does NOT matter (e.g. picking a 3-fruit salad). In permutations (nPr), order DOES matter (e.g. setting a 3-digit safe code).
Why is 10C3 equal to 10C7?
Because combinations are perfectly symmetric: nCr = nC(n-r). Choosing 3 items to include out of 10 is mathematically identical to choosing 7 items to exclude.
How many combinations of 5 cards can be drawn from a 52-card deck?
There are exactly 2,598,960 possible 5-card poker hands (52C5 = 52! ÷ [5! · 47!] = 2,598,960).