Combination Calculator

About Combination Calculator

Welcome to the Combination Calculator, a comprehensive tool for calculating combinations C(n,k) with step-by-step solutions, Pascal's Triangle visualization, and interactive diagrams. Whether you are solving probability problems, studying combinatorics, calculating lottery odds, or working on counting problems, this calculator provides detailed explanations and visual representations to help you understand the mathematics behind combinations.

What is a Combination?

A combination is a selection of items from a larger set where the order of selection does not matter. It answers the question: "In how many ways can I choose k items from n items?"

For example, if you want to choose 3 students from a class of 10 to form a committee, the combination C(10,3) = 120 tells you there are 120 different possible committees. The order in which you select the students doesn't matter - selecting Alice, Bob, then Carol gives the same committee as selecting Carol, Alice, then Bob.

The Combination Formula

Where:

• n = Total number of items in the set
• k = Number of items to choose
• n! = Factorial of n (product of all positive integers from 1 to n)
• C(n,k) = Number of possible combinations (also written as _nC_k or "n choose k")

Combinations vs Permutations

The key difference between combinations and permutations is whether order matters:

For the same n and k values, permutations always yield larger results because they count each group multiple times (once for each possible ordering).

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 triangle provides a visual way to find combination values:

• Row n contains all values C(n, 0), C(n, 1), ..., C(n, n)
• The first and last numbers in each row are always 1
• C(n, k) = C(n, n-k) - the triangle is symmetric

For example, row 5 of Pascal's Triangle shows: 1, 5, 10, 10, 5, 1, corresponding to C(5,0), C(5,1), C(5,2), C(5,3), C(5,4), C(5,5).

How to Use This Calculator

1. Enter n (total items): Input the total number of items in your set. The maximum value is 170.
2. Enter k (items to choose): Input how many items you want to select. This must be less than or equal to n.
3. Click Calculate: The calculator will compute C(n,k) and display:
   • The final result with thousands separators for readability
   • Step-by-step calculation breakdown
   • Pascal's Triangle visualization (for n ≤ 12)
   • All possible combinations listed (for small results)
   • Real-world application examples
4. Try preset values: Use the quick preset buttons to explore common combination problems.

Real-World Applications

Lottery and Gambling

Combinations are essential for calculating lottery odds. For a 6/49 lottery (choosing 6 numbers from 49), C(49,6) = 13,983,816 possible combinations, giving odds of about 1 in 14 million.

Probability and Statistics

The binomial probability formula uses combinations: P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where p is the probability of success in a single trial.

Team Selection

When selecting a committee of 5 from 20 candidates, C(20,5) = 15,504 possible committees can be formed.

Card Games

Poker hand probabilities rely on combinations. A standard deck has C(52,5) = 2,598,960 possible 5-card hands.

Handshake Problems

If n people shake hands with everyone exactly once, the total number of handshakes is C(n,2) = n(n-1)/2.

Important Properties of Combinations

Symmetry Property

Choosing k items to include is equivalent to choosing (n-k) items to exclude.

Pascal's Identity

This recursive relationship is why Pascal's Triangle works - each number is the sum of the two above it.

Sum of Row

The sum of all combinations in row n equals 2^n, representing all possible subsets of an n-element set.

Frequently Asked Questions

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. It is denoted as C(n,k) or "n choose k", representing the number of ways to choose k items from n items. Unlike permutations, combinations treat {A,B,C} and {C,B,A} as the same selection.

What is the formula for combinations?

The combination formula is C(n,k) = n! / (k! × (n-k)!), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. This formula calculates how many different groups of k items can be selected from n items without considering order.

What is the difference between combinations and permutations?

The key difference is order: in combinations, order does not matter (selecting A,B,C is the same as C,B,A), while in permutations, order matters (ABC and CBA are different arrangements). Combinations count groups, permutations count arrangements.

What is Pascal's Triangle and how does it relate to combinations?

Pascal's Triangle is a triangular array 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). This provides a visual way to find combination values without calculation.

What are real-world applications of combinations?

Combinations have many practical applications: calculating lottery odds, counting handshakes at a party, determining poker hand probabilities, selecting team members from a group, and solving problems in probability, statistics, and computer science.

Related Calculators

• Permutation Calculator - Calculate P(n,r) when order matters
• Factorial Calculator - Calculate n! for any number
• Binomial Distribution Calculator - Calculate binomial probabilities
• nCr Calculator - Another notation for combinations

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