Combination Calculator | nCr Calculator & Combinations Formula

📖 How to Use This Combination Calculator

Enter total items (n) - The total number of distinct items available to choose from.

Enter chosen items (r) - How many items you want to select.

Click "Calculate nCr" - Get the number of possible combinations with step-by-step explanation.

🔢 Understanding Combinations (nCr)

A combination is a way of selecting items from a larger set where the order of selection does NOT matter. For example, choosing 3 fruits from a basket of 10 - the set {apple, banana, orange} is the same combination regardless of which fruit was picked first.

The notation C(n,r) or "n choose r" represents the number of combinations possible. This is different from permutations (nPr) where order matters.

🧮 Combinations Formula

C(n,r) = n! / (r! × (n-r)!)

Where:
• n! = n × (n-1) × (n-2) × ... × 1 (factorial)
• r ≤ n (can't choose more than available)
• For r > n, C(n,r) = 0

📊 Real-World Examples of Combinations

Lottery Tickets: Choosing 6 numbers from 49 (C(49,6) ≈ 13,983,816 possibilities)
Poker Hands: Number of possible 5-card hands from a 52-card deck (C(52,5) = 2,598,960)
Committee Selection: Choosing 3 people from a group of 10 for a committee
Ice Cream Flavors: Choosing 2 flavors from 12 available options

💡 Key Properties of Combinations

C(n,r) = C(n, n-r) - Choosing r items is the same as leaving out n-r items
C(n,0) = 1 - There's exactly one way to choose nothing
C(n,1) = n - There are n ways to choose 1 item
C(n,n) = 1 - There's exactly one way to choose all items
C(n,r) = C(n-1,r-1) + C(n-1,r) - Pascal's Rule

❓ Frequently Asked Questions (FAQ)

What is the difference between combinations and permutations?

In combinations, order does NOT matter. {A,B,C} is the same as {C,B,A}. In permutations, order matters. For example, choosing a president, vice president, and secretary from a group - each arrangement is different.

What does "n choose r" mean?

"n choose r" (written as C(n,r) or nCr) is the number of ways to choose r items from a set of n items where order doesn't matter. It's also known as the binomial coefficient.

Can r be greater than n?

No, you cannot choose more items than are available. If r > n, the result is 0 because there are no possible combinations.

What is the largest value this calculator can handle?

This calculator can handle large numbers up to approximately 1 quadrillion (10^15). For very large results, scientific notation is used (e.g., 1.23e+15).

How is C(n,r) related to Pascal's Triangle?

Pascal's Triangle is a triangular array where each number is the sum of the two numbers above it. Each number in the triangle represents C(n,r) where n is the row number and r is the position in that row.

What are binomial coefficients used for?

Binomial coefficients appear in the binomial theorem: (x + y)^n = Σ C(n,k) × x^(n-k) × y^k. They're also used in probability, combinatorics, and algebra for expanding binomials.