Calculating Combinations - Tutorials

Introduction

This practical tutorial teaches how to calculate combinations and permutations, fundamental for understanding lottery probabilities. You will learn combinatorial formulas and apply them in practical scenarios.

Fundamental Concepts

Before calculating, understand the difference between combinations and permutations:

Combination

Order does NOT matter

Example: {1, 2, 3} is the same as {3, 2, 1}

Permutation

Order matters

Example: 1-2-3 is different from 3-2-1

Step 1: Calculate Combinations

Combinations are used when order does not matter, as in lotteries where {1, 2, 3} = {3, 2, 1}.

Combination Formula

C(n,k) = n! / (k! × (n-k)!)

• n: Total elements available
• k: Number of elements to choose
• ! (factorial): Product from 1 to n

Step 2: Example - Mega-Sena

Mega-Sena: we choose 6 numbers from 60:

Step by Step Calculation

C(60, 6) = 60! / (6! × 54!)

Simplificando os fatoriais:
C(60, 6) = (60 × 59 × 58 × 57 × 56 × 55) / (6 × 5 × 4 × 3 × 2 × 1)

C(60, 6) = 50.063.860 combinações possíveis

C(60,6) = 50,063,860 combinations. Probability of matching all 6: 1 in 50,063,860.

Step 3: Other Lotteries

Apply the same formula:

Quina

5 numbers from 80:

C(80,5) = 24,040,016

Lotofácil

15 numbers from 25:

C(25,15) = 3,268,760

Step 4: Using a Calculator

For manual calculations, simplify: multiply the k largest of n and divide by k!

Practical Tips

✅ Best Practices

• Use scientific or online calculator for large numbers
• Check if order matters (combination vs. permutation)
• Simplify factorials by canceling common terms
• Verify with known examples (Mega-Sena = 50,063,860)

Practical Applications

Calculating combinations is useful for:

• Calculating probabilities and real odds
• Comparing lotteries
• System analysis and coverage
• Understanding combinatorial analysis

Conclusion

Calculating combinations is fundamental to understanding lottery probabilities. With practice, you will calculate quickly in any scenario.

💡 Next Steps

Explore how to use combinations to calculate probabilities and understand the odds in different games.

🎯 Calculating Combinations