📝 Practice Worksheets
🔑 Key Concepts
- A combination is used when order does not matter.
- The notation \( nCr \) means choosing \( r \) items from \( n \) total items.
- The full formula is \( nCr = \frac{n!}{r!(n-r)!} \).
- The shortcut is to multiply the first \( r \) factors from \( n! \), then divide by \( r! \).
- Combinations are useful for groups, teams, committees, toppings, letters, and other “choose” situations.
✏️ Worked Examples
🧠 Math Vocabulary
- Combination: A selection of items where order does not matter.
- Permutation: An arrangement of items where order does matter.
- Factorial: The product of a whole number and all positive whole numbers below it, such as \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
- nCr: A notation used to show the number of combinations of \( r \) items chosen from \( n \) total items.
- n: The total number of items available.
- r: The number of items being chosen.
- Order Does Not Matter: The same group counts once, even if the items are listed in a different order.
💡 Main Idea
Combinations help count how many groups can be made when order does not matter. The nCr shortcut works because the full factorial formula cancels down to the first few factors of \( n! \) divided by \( r! \).
📚 What You Should Already Know
Students should know how to multiply whole numbers, simplify fractions, understand factorial notation, and decide whether order matters in a counting situation.
🚀 What Comes Next
Next, students can compare combinations with permutations and apply counting strategies to probability problems involving sample spaces.
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