Test Questions for Permutations with Repetition and Duplicates

📚 Quick Study Guide

• 🔢 Permutations with Repetition: When you have $n$ objects and want to arrange $r$ of them, where repetition is allowed, the number of permutations is $n^r$.
• 👯 Permutations with Duplicates: If you have $n$ objects with $n_1$ of one kind, $n_2$ of another kind, and so on, up to $n_k$ of the $k$-th kind, the number of distinct permutations is given by: $\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$, where $n = n_1 + n_2 + ... + n_k$.
• 💡 Key Differences: Repetition allows you to use an item more than once in your arrangement. Duplicates deal with items that are inherently identical within a set.
• 📝 Example - Repetition: Forming a 3-digit PIN using digits 0-9 (repetition allowed) falls under permutations with repetition ($10^3 = 1000$ possibilities).
• 📌 Example - Duplicates: Arranging the letters in the word 'APPLE' falls under permutations with duplicates (5 letters, with 'P' repeated twice: $\frac{5!}{2!} = 60$ possibilities).

📝 Practice Quiz

1. What is the number of different 4-digit PINs that can be formed using the digits 0-9, where repetition is allowed?
   1. 1000
   2. 10000
   3. 5040
   4. 720
2. How many distinct permutations are there of the letters in the word 'BANANA'?
   1. 60
   2. 120
   3. 360
   4. 720
3. A restaurant offers 5 different toppings for a pizza. How many different pizzas can be made if you can choose any number of toppings (including no toppings), and you can choose the same topping multiple times?
   1. 25
   2. 32
   3. 120
   4. 240
4. How many different 5-letter words (real or imaginary) can be formed from the letters A, B, C, D, and E, if repetition of letters is allowed?
   1. 120
   2. 625
   3. 3125
   4. 7776
5. A bag contains 3 red balls, 2 blue balls, and 1 green ball. How many different ways can you arrange these balls in a row?
   1. 6
   2. 30
   3. 60
   4. 720
6. How many different signals can be made by arranging 3 red flags, 4 white flags, and 2 blue flags in a row?
   1. 1260
   2. 15120
   3. 30240
   4. 362880
7. You have 4 different types of donuts. How many ways can you select 6 donuts, allowing repetition?
   1. 84
   2. 210
   3. 720
   4. 4096