Problem 11 - Algebra and Trigonometry

Question

7–12 Find the number of distinguishable permutations of the given letters. $$X X Y Y Y Z Z Z$$

Step-by-Step Solution

Answer

There are 560 distinguishable permutations.

Step 1: Understanding the Problem

The problem involves finding the number of distinguishable permutations of the given multiset of letters: \(X X Y Y Y Z Z Z\). We need to consider the fact that some letters repeat.

Step 2: Formula for Permutations of Multisets

The formula to find the number of distinguishable permutations of a multiset is given by: \[\frac{n!}{n_1! \times n_2! \times \, ... \, \times n_k!}\]where \(n\) is the total number of elements, and \(n_1, n_2, ..., n_k\) are the frequencies of each distinct element.

Step 3: Calculate Total Elements and Frequencies

In this problem, the total number of elements \(n\) is 8, as we have the letters \(X\), \(Y\), and \(Z\) arranged as: 2 X's, 3 Y's, and 3 Z's. Thus, \(n_X = 2\), \(n_Y = 3\), and \(n_Z = 3\).

Step 4: Apply the Formula

Substitute the values into the formula:\[\frac{8!}{2! \times 3! \times 3!}\]Calculate \(8!\), \(2!\), \(3!\) and simplify.

Step 5: Calculate Factorials

Calculate the individual factorials:\(8! = 40,320\)\(2! = 2\)\(3! = 6\)

Step 6: Simplify the Expression

Substitute the factorial values:\[\frac{40,320}{2 \times 6 \times 6} = \frac{40,320}{72} = 560\]

Step 7: Conclusion

The number of distinguishable permutations of the given letters \(X X Y Y Y Z Z Z\) is 560.

Key Concepts

Factorials

Factorials are core elements in the understanding of permutations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For example, if you have a number 5, its factorial is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

In permutations, factorials help calculate the number of ways objects can be arranged. The larger the number, the more complex its factorial can become. However, the basic principle remains simple: multiply all whole numbers leading up to the number in question. Factorials decrease rapidly as you break down to smaller numbers. This characteristic is useful when simplifying permutation formulas.

Multisets

A multiset is a collection of elements similar to a set, but with a key difference: elements can repeat. This repetition is crucial when dealing with permutations because it affects how we calculate arrangements.
In our problem with the letters

• X X Y Y Y Z Z Z

we have a multiset consisting of letters that repeat different numbers of times. This repetition must be accounted for to determine distinguishable permutations accurately. By identifying and noting the frequency of each unique letter in the multiset, we use these amounts in our formula to adjust for repeated arrangements. This altered count affects the total number of unique permutations.

Permutation Formula

The permutation formula for multisets is a helpful tool to determine how objects with repeated elements can be rearranged. The formula is represented as:

• \[\frac{n!}{n_1! \times n_2! \times \, ... \, \times n_k!}\]

where \(n\) is the total number of elements, and \(n_1, n_2, ..., n_k\) represent the frequencies of each unique element.
This formula helps eliminate the overcounting issue of repeated elements, adjusting the factorial for repeated items by dividing the total factorial \(n!\) by the factorials of the frequencies. The result is a count of only unique arrangements, making it integral to solving permutation problems involving multisets.

Distinguishable Permutations

Distinguishable permutations refer to the different ways items can be arranged when some items are identical. This concept is particularly useful in computing arrangements for words or objects that contain repeated elements.
For example, when arranging the multiset

• X X Y Y Y Z Z Z

we apply the permutation formula to account for the repeated X's, Y's, and Z's. Upon calculating, we find 560 unique sequences of these letters.
Understanding distinguishable permutations helps to solve problems in combinatorics accurately, allowing you to account for potential redundancy in arrangements. It's all about identifying the variations that make an arrangement unique.