Combination Calculator

A combination is a selection of items from a collection, such that the order of selection does not matter. In other words, it's a way to calculate the number of possible ways to select r items from a set of n items, regardless of the order. The formula for combinations is: C(n,r) = n! / (r! × (n-r)!), where n! represents the factorial of n.

The key difference is whether the order of selection matters:

• Combinations: Order doesn't matter (e.g., selecting a team of 3 people from 10 candidates)
• Permutations: Order matters (e.g., ranking top 3 finishers in a race with 10 participants)

Mathematically, the permutation formula includes all possible arrangements of the selected items, while the combination formula does not. Permutation formula: P(n,r) = n! / (n-r)!

To calculate combinations by hand:

1. Identify n (total number of items) and r (number of items to select)
2. Calculate n! (n factorial): n × (n-1) × (n-2) × ... × 2 × 1
3. Calculate r! (r factorial): r × (r-1) × (r-2) × ... × 2 × 1
4. Calculate (n-r)! in the same way
5. Apply the formula: C(n,r) = n! / (r! × (n-r)!)

For example, to calculate C(5,3): 5! / (3! × (5-3)!) = 120 / (6 × 2) = 120 / 12 = 10

• Lottery and gambling: Calculating odds of winning (e.g., picking 6 numbers from 49)
• Sports team selection: Number of ways to select a team from a pool of players
• Card games: Calculating poker hand probabilities
• Statistical sampling: Determining possible sample combinations
• Molecular biology: Analyzing possible genetic combinations
• Committee formation: Calculating ways to form committees from a larger group
• Cryptography: Analyzing possible combinations for security systems

Factorials are used in the combination formula because they efficiently count the number of ways to arrange items. The numerator n! represents all possible arrangements of n items. Since in combinations the order doesn't matter, we divide by r! to remove the redundant arrangements of the r selected items. We also divide by (n-r)! to remove redundant arrangements of the unselected items. This gives us exactly the number of unique ways to select r items from n items, regardless of order.

Combinations have several important properties:

• Symmetry: C(n,r) = C(n,n-r) - selecting r items is the same as not selecting (n-r) items
• Pascal's Identity: C(n,r) = C(n-1,r-1) + C(n-1,r) - used in Pascal's Triangle
• Special Cases: C(n,0) = C(n,n) = 1 - there's exactly one way to select none or all items
• Sum of Row: C(n,0) + C(n,1) + ... + C(n,n) = 2^n - total number of possible subsets

This calculator can handle values of n up to 170. Beyond this, factorial calculations exceed JavaScript's maximum safe integer limits. For most practical applications, this limit is more than sufficient. If you need to calculate combinations with extremely large numbers, specialized mathematical software with arbitrary-precision arithmetic would be required. Note that even for moderate values of n, the number of combinations can become extremely large.