Permutation of n things, p are alike, q are alike, r are alike

When dealing with permutations, we often encounter situations where certain items are indistinguishable from one another. In such cases, the formula for permutations needs to be adjusted to account for these repetitions. Let's explore the concept of permutations when there are groups of identical items.

Understanding Permutations with Identical Items

A permutation is an arrangement of items in a specific order. When all items are distinct, the number of permutations of n items is n! (n factorial). However, when there are groups of identical items, the number of unique permutations is reduced.

The General Formula

The general formula for the number of permutations of n items, where there are groups of identical items, is given by:

[ P = \frac{n!}{p! \cdot q! \cdot r! \cdot \ldots} ]

Where:

• n! is the factorial of the total number of items.
• p!, q!, r!, ... are the factorials of the number of identical items in each group.

This formula accounts for the fact that permuting identical items within their respective groups does not result in a distinct arrangement.

Table of Differences and Important Points

Aspect	Distinct Items	Identical Items
Formula	n!	\frac{n!}{p! \cdot q! \cdot r! \cdot \ldots}
Explanation	Each item is unique, so all possible orders are counted.	Identical items in a group can be rearranged without creating a new permutation.
Example	Arranging 3 distinct books (A, B, C)	Arranging 3 books where 2 are identical (A, B, B)
Number of Permutations	3! = 6	\frac{3!}{2!} = 3

Examples to Explain Important Points

Example 1: Basic Scenario

Suppose we have 5 balls, 3 of which are red (indistinguishable from each other), and 2 of which are blue (also indistinguishable from each other). To find the number of unique arrangements (permutations) of these balls, we use the formula:

[ P = \frac{n!}{p! \cdot q!} ]

Here, n = 5 (total balls), p = 3 (red balls), and q = 2 (blue balls).

[ P = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 ]

So, there are 10 unique ways to arrange these balls.

Example 2: More Complex Scenario

Imagine we have a collection of 10 fruits: 4 apples, 3 bananas, and 3 oranges, which are all identical within their respective types. To find the number of unique arrangements, we apply the formula:

[ P = \frac{n!}{p! \cdot q! \cdot r!} ]

Here, n = 10 (total fruits), p = 4 (apples), q = 3 (bananas), and r = 3 (oranges).

[ P = \frac{10!}{4! \cdot 3! \cdot 3!} = \frac{3,628,800}{24 \cdot 6 \cdot 6} = \frac{3,628,800}{864} = 4,200 ]

Thus, there are 4,200 unique ways to arrange these fruits.

Conclusion

When calculating permutations with identical items, it is crucial to divide by the factorials of the number of identical items to avoid overcounting arrangements. This ensures that only distinct permutations are considered. Understanding this concept is essential for solving problems in combinatorics and probability where items are not all unique.