Real-World Examples of Permutations with Identical Items

📚 Quick Study Guide

• 🔢 Permutations with identical items involve arranging a set of $n$ objects where some are indistinguishable.
• 📝 The formula to calculate the number of permutations is: $\frac{n!}{n_1! * n_2! * ... * n_k!}$, where $n$ is the total number of items, and $n_1, n_2, ..., n_k$ are the counts of each group of identical items.
• 💡 Remember that $n!$ (n factorial) means $n * (n-1) * (n-2) * ... * 2 * 1$.
• 🧮 This formula adjusts for overcounting that occurs when identical items are swapped.
• 🔑 Key application: Arranging letters in a word with repeated letters, arranging colored balls, etc.

Practice Quiz

1. What is the number of distinct permutations of the letters in the word "STATISTICS"?
   1. 50,400
   2. 453,600
   3. 60,480
   4. 362,880
2. How many different ways can you arrange 3 red beads, 2 blue beads, and 1 green bead on a string?
   1. 720
   2. 60
   3. 120
   4. 30
3. In how many ways can the letters of the word "BANANA" be arranged so that the two N's are always together?
   1. 60
   2. 120
   3. 24
   4. 360
4. A shelf contains 5 identical math books, 3 identical science books, and 2 identical history books. How many different ways can the books be arranged on the shelf?
   1. 15,120
   2. 2,520
   3. 30,240
   4. 3,628,800
5. How many distinct arrangements are there of the letters in the word "MISSISSIPPI"?
   1. 34,650
   2. 3,628,800
   3. 346,500
   4. 1,260
6. A bag contains 4 identical white balls and 3 identical black balls. In how many ways can 4 balls be drawn from the bag?
   1. 35
   2. 7
   3. 1
   4. 4
7. Find the number of permutations of the letters of the word 'ENGINEERING'.
   1. 277,200
   2. 3,991,6800
   3. 11!
   4. 166,3200