Factorials vs. combinations explained for Pre-Calculus students

📚 Factorials vs. Combinations: Unlocked!

Alright, let's break down factorials and combinations. They're both about counting possibilities, but they differ in a crucial way: whether the order matters.

🧮 Definition of Factorial

A factorial (denoted by !) calculates the product of all positive integers less than or equal to a given number. It answers the question: "In how many ways can I arrange this many distinct items?"

• 🔢 The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$.
• ✍️ Formula: $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$
• 📌 Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. This means there are 120 ways to arrange 5 distinct objects.

🎲 Definition of Combination

A combination, on the other hand, counts the number of ways to choose a subset of items from a larger set, where the order of selection doesn't matter. Think of picking a team – the order you pick the players doesn't change the team itself.

• 🎁 A combination is a selection of items from a set where the order of selection does not matter.
• 🧪 Formula: The number of combinations of choosing $r$ items from a set of $n$ items is given by: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$
• 🌍 Example: Suppose you have 5 friends, and you want to invite 3 of them to a party. The number of different groups of 3 friends you can invite is $\binom{5}{3} = \frac{5!}{3!2!} = 10$.

🆚 Factorial vs. Combination: The Ultimate Comparison

🔑 Key Takeaways

• 💡 Factorials are used when the order of items is important (permutations).
• ✔️ Combinations are used when the order of items is not important.
• 📝 Both concepts are fundamental in probability and counting problems.
• 🧠 Understanding the difference helps in solving a wide range of mathematical problems.