Combination Calculator
Calculate combinations C(n, r) where order does not matter, combinations with repetition (multisets), Pascal's symmetry, and permutations P(n, r).
nCr Combination Solver
Enter total items (n) and chosen items (r).
Combination Output
Quick Summary
Our combination calculator ncr order doesn't matter steps tool calculates the number of ways to choose $r$ items from $n$ items where selection order is irrelevant. It computes standard binomial coefficients ($\binom{n}{r}$), combinations with repetition ($\binom{r+n-1}{r}$), and displays Pascal's triangle symmetry ($C(n, r) = C(n, n-r)$).
How It Works: Unordered Selection & Binomial Coefficients
In combinatorics, combinations count subsets where ORDER DOES NOT MATTER:
1. **Without Repetition ($C(n,r)$):** Divide permutations $P(n,r)$ by $r! \implies C(n,r) = \binom{n}{r} = \n! / r!(n-r)!$.
2. **With Repetition ($\binom{r+n-1}{r}$):** Uses the Stars and Bars theorem to count multisets: $\(r+n-1)! / r!(n-1)!$.
3. **Pascal's Triangle Symmetry:** Choosing $r$ items from $n$ is equivalent to choosing which $n-r$ items to leave behind $\implies C(n,r) = C(n, n-r)$.
Formula Explanation
Your combination calculations use the binomial coefficient formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for selecting a 4-person committee from 10 members ($n=10, r=4$):
- Step 1 (Identify Parameters): Total members $n = 10$, committee size $r = 4$.
- Step 2 (Calculate n!): $10! = \mathbf{3,628,800}$.
- Step 3 (Calculate r! and (n-r)!): $4! = \mathbf{24}$, $(10-4)! = 6! = \mathbf{720}$.
- Step 4 (Multiply Denominator): $r! \times (n-r)! = 24 \times 720 = \mathbf{17,280}$.
- Step 5 (Divide Factorials): $C(10, 4) = \3,628,800 / 17,280 = \mathbf{210}$, delivering **Combinations C(10,4) = 210**!
Calculation Examples: Real-World Scenario Comparison
Compare combination results across committees, lotteries, card hands, and ice cream flavor selections:
| Real-World Scenario | Total (n) | Chosen (r) | Combinations C(n, r) | With Repetition | Equivalent Permutation P(n, r) |
|---|---|---|---|---|---|
| 4-person committee from 10 members | 10 | 4 | 210 | 715 | 5,040 |
| Lottery: 6 numbers from 49 | 49 | 6 | 13,983,816 | 25,827,165 | 10,068,347,520 |
| Poker: 5-card hand from 52 cards | 52 | 5 | 2,598,960 | 3,819,816 | 311,875,200 |
| 3 ice cream scoops from 5 flavors | 5 | 3 | 10 (Unique) | 35 (Repeated) | 60 |
Benefits of Using the Combination Calculator
Utilizing this calculator provides essential statistical, gaming, and business team building benefits:
- Lottery & Gambling Odds: Calculates exact winning odds for lotteries (e.g. 1 in 13.98 million for 6/49 lotto).
- Team & Committee Formation: Computes possible team compositions without duplicate orderings.
- Pascal's Symmetry Verification: Confirms $C(n, r) = C(n, n-r)$ for binomial expansion coefficients.
- Multiset & Stars and Bars Solver: Computes combinations with repetition for inventory and physics state distributions.
Frequently Asked Questions (FAQ)
What is a combination in mathematics?
A combination is a selection of items from a larger set where the order of selection does NOT matter.
What is the nCr formula?
C(n, r) = n! / (r! × (n - r)!), where n is total items and r is items chosen.
Why is C(n, r) smaller than P(n, r)?
Because permutations count every unique ordering as distinct, whereas combinations group all equivalent orderings into a single group by dividing by r!.
What is Pascal's symmetry property?
Pascal's symmetry states that C(n, r) = C(n, n - r) (e.g. choosing 4 items from 10 is the same as choosing 6 items to leave out: C(10, 4) = C(10, 6) = 210).
What is C(n, 0) and C(n, n)?
C(n, 0) = 1 and C(n, n) = 1 (there is exactly 1 way to choose 0 items or all n items).
How do you calculate combinations with repetition?
Use the formula C(r + n - 1, r) = (r + n - 1)! / (r! × (n - 1)!).
What are the odds of winning a 6/49 lottery?
The total number of 6-number combinations from 49 is C(49, 6) = 13,983,816, giving odds of 1 in 13,983,816.
How many 5-card poker hands are possible in a 52-card deck?
C(52, 5) = 2,598,960 possible hands.
What is C(10, 4)?
C(10, 4) = 10! / (4! × 6!) = 3,628,800 / (24 × 720) = 210.
Is order important when choosing a team of people?
No! Since selecting Alice, Bob, and Charlie results in the same team as Charlie, Bob, and Alice, you use combinations.