Combination Calculator
Calculate combinations ($C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}$), combinations with repetition ($\binom{n+r-1}{r}$), and Pascal's triangle coefficients.
Calculate Unordered Combinations C(n, r)
Enter total available items (n) and number of chosen items (r).
Calculation Results
Calculated using combination binomial coefficient equations: C(n, r) = \binom{n}{r} = \frac{n!}{r! (n - r)!}
Quick Summary
The Combination Calculator evaluates unordered selections $C(n, r) = \frac{n!}{r!(n - r)!}$, combinations with repetition ($\binom{n+r-1}{r}$), and Pascal's triangle coefficients.
Formula Explanation
C(n, r) = \binom{n}{r} = \frac{n!}{r! (n - r)!}
\text{Combinations with Repetition} = \binom{n + r - 1}{r}How It Works
A combination is a selection of items from a set where order does not matter. The Combination Calculator divides permutation count $P(n, r)$ by $r!$ to eliminate duplicate orderings.
Step-by-Step Worked Example
Practical Problem: Calculate the number of combinations when choosing $r = 3$ committee members from $n = 5$ candidates.
- Step 1: Calculate $n! = 5!$: $5! = \mathbf{120}$.
- Step 2: Calculate $r! = 3!$: $3! = \mathbf{6}$.
- Step 3: Calculate $(n - r)! = 2!$: $2! = \mathbf{2}$.
- Step 4: Divide $n!$ by $(r! \times (n-r)!)$: $C(5, 3) = 120 / (6 \times 2) = 120 / 12 = \mathbf{10\text{ combinations}}$.
- Step 5: Interpretation: There are 10 unique 3-member committees that can be formed from 5 candidates.
Real-World Calculation Examples
Scenario 1: 3 Items from 5 (C(5, 3))
Parameters: n = 5, r = 3
Result: C(5, 3) = 10 combinations (Order irrelevant).
Scenario 2: Lottery Ticket Ball Selection (6 from 49)
Parameters: n = 49, r = 6
Result: C(49, 6) = 13,983,816 combinations (1 in 14M odds).
Scenario 3: Poker Hands (5 cards from 52)
Parameters: n = 52, r = 5
Result: C(52, 5) = 2,598,960 possible hands.
Scenario 4: Ice Cream Scoops with Repetition (3 scoops, 5 flavors)
Parameters: n = 5, r = 3 with repetition
Result: C(5+3-1, 3) = C(7, 3) = 35 combinations.
Key Benefits of Using This Calculator
Unordered Combinations $C(n, r)$
Computes exact unordered subset selections where item order is irrelevant.
Pascal's Triangle Coefficient
Displays binomial expansion coefficients $(a+b)^n$.
Combinations with Repetition
Outputs combinations with replacement $\binom{n+r-1}{r}$ (stars and bars theorem).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a combination?
A combination is a selection of items from a collection where the order of selection does not matter ($C(n, r) = \frac{n!}{r!(n - r)!}$).
What is symmetry property of combinations?
$C(n, r) = C(n, n - r)$ (e.g. choosing 3 items to keep is identical to choosing 2 items to leave behind out of 5).
How do I calculate combinations in Excel?
Use formula =COMBIN(n, r) for no repetition or =COMBINA(n, r) for repetition allowed.
What is Pascal's Identity?
Pascal's Identity states $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$.
What is sum of a row in Pascal's Triangle?
The sum of row $n$ in Pascal's Triangle is $\sum_{r=0}^{n} \binom{n}{r} = 2^n$.
What is Binomial Theorem?
$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$.
What is $C(n, 0)$ and $C(n, n)$ equal to?
$C(n, 0) = 1$ and $C(n, n) = 1$.
What is Stars and Bars theorem?
Stars and Bars theorem calculates combinations with repetition: $\binom{n + r - 1}{r}$.
Why is lottery probability calculated using combinations?
Because lottery numbers can be drawn in any order, combinations $C(N, r)$ determine winning ticket probabilities.
What is Vandermonde's Identity?
$\binom{m + n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r - k}$.