Real-world examples of combinations in everyday life

📚 Quick Study Guide

🔢 Combination: A selection of items where the order doesn't matter. ➗ Formula: The number of combinations of $n$ items taken $r$ at a time is given by: $C(n, r) = \frac{n!}{r!(n-r)!}$, where $n!$ (n factorial) is the product of all positive integers up to $n$. 🗓️ Key Concept: The difference between permutations and combinations is that order matters in permutations but not in combinations. 💡 Tip: Use combinations when you're selecting groups and the arrangement within the group is irrelevant.

🧪 Practice Quiz

1. You're ordering a pizza with 3 toppings. If there are 8 available toppings, how many different pizza combinations are possible?
   1. 24
   2. 56
   3. 336
   4. 6720
2. A committee of 5 people needs to be formed from a group of 10. How many different committees can be formed?
   1. 50
   2. 252
   3. 30240
   4. 100000
3. You have 6 different books and want to choose 2 to take on a trip. How many different sets of books can you choose?
   1. 12
   2. 15
   3. 30
   4. 360
4. A bag contains 4 red balls and 5 blue balls. You randomly pick 3 balls. How many combinations result in exactly 2 blue balls and 1 red ball?
   1. 6
   2. 30
   3. 40
   4. 120
5. In a card game, you're dealt 5 cards from a standard deck of 52. How many different hands are possible?
   1. 2598960
   2. 311875200
   3. 120
   4. 52
6. A bakery offers 7 different types of donuts. You want to buy a box of 4 donuts. If you can choose the same type multiple times, how many different combinations are possible?
   1. 28
   2. 35
   3. 210
   4. 840
7. You are creating a salad with 4 ingredients from a selection of 10. How many different salads can you make?
   1. 40
   2. 210
   3. 5040
   4. 3628800