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Key Combination Facts
- C(n, 0) = 1 — there is exactly one way to choose nothing
- C(n, n) = 1 — there is exactly one way to choose everything
- C(n, 1) = n — choosing one item from n gives n options
- C(n, r) = C(n, n-r) — choosing r items is equivalent to choosing which n-r to leave out
- C(n, r) = 0 when r > n — you cannot choose more items than exist
The Combination Calculator on TechConverter.me handles any values of n and r instantly, including large numbers that would overflow standard calculators. Enter your values and get the result along with the step-by-step calculation.
Examples
Example 1: Basic Combination Calculation
How many ways can you choose 3 people from a team of 10 to form a committee?
C(10, 3) = 10! / (3! × 7!)
= (10 × 9 × 8) / (3 × 2 × 1)
= 720 / 6
= 120
There are 120 different possible committees. Order does not matter — a committee of Alice, Bob, and Carol is the same regardless of who was selected first. If order mattered (e.g., choosing a president, vice president, and secretary), you would use permutations instead: P(10, 3) = 720.
Example 2: Lottery Probability
A lottery requires players to choose 6 numbers from 1 to 49. How many possible combinations are there?
C(49, 6) = 49! / (6! × 43!)
= (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
= 10,068,347,520 / 720
= 13,983,816
There are nearly 14 million possible combinations, meaning the probability of winning with a single ticket is approximately 1 in 13,983,816 (about 0.0000071%). This is why lottery jackpots can grow so large before being won.
Example 3: Poker Hand Probabilities
A standard poker hand consists of 5 cards from a 52-card deck. How many possible 5-card hands are there?
Total hands: C(52, 5) = 2,598,960
Specific hand counts:
Royal flush: C(4, 1) = 4
Straight flush: 36 (9 straights × 4 suits, excluding royal)
Four of a kind: C(13, 1) × C(4, 4) × C(48, 1) = 624
Full house: C(13, 1) × C(4, 3) × C(12, 1) × C(4, 2) = 3,744
Flush: C(13, 5) × 4 - 40 = 5,108
Probability of a flush: 5,108 / 2,598,960 ≈ 0.197%
Combination calculations are the foundation of all poker probability analysis, helping players understand the relative rarity of different hands.