Problem 8 - Algebra and Trigonometry

Question

7–12 Find the number of distinguishable permutations of the given letters. $$A A A B B B C C C$$

Step-by-Step Solution

Answer

There are 1680 distinguishable permutations of the letters A, A, A, B, B, B, C, C, C.

Step 1: Understand the Problem

We need to find the number of distinguishable permutations for the given letters: A, A, A, B, B, B, C, C, C. This means determining how many unique ways we can arrange these letters.

Step 2: Identify the Total Number of Letters

Count the total number of letters, which includes A, B, and C. In this case, there are 9 letters in total: 3 A's, 3 B's, and 3 C's.

Step 3: Apply the Permutation Formula with Repeated Elements

Using the formula for permutations of multiset, the number of distinguishable permutations is given by \[ \frac{n!}{n_1! \times n_2! \times \, ... \, \times n_k!} \] where \(n\) is the total number of items, and \(n_1, n_2, ..., n_k\) are the frequencies of the distinct items.

Step 4: Substitute the Values into the Formula

Here, the total number of letters \( n = 9 \), and the frequencies are \(3\) for A's, \(3\) for B's, and \(3\) for C's. So the formula becomes: \[ \frac{9!}{3! \times 3! \times 3!} \]

Step 5: Calculate Factorials

Calculate each factorial. \(9! = 362880\), \(3! = 6\).

Step 6: Solve the Permutation Expression

Substitute the factorials into the expression: \[ \frac{362880}{6 \times 6 \times 6} = \frac{362880}{216} = 1680 \]

Step 7: Verify the Solution

Ensure that calculations are correct. Confirm that the final result is consistent with the principles of permutations with repeated elements.

Key Concepts

Permutation Formula

When we talk about permutations, we're discussing the arrangements of objects. The permutation formula is especially important when calculating the number of ways to arrange items. For distinguishable permutations involving repeated elements, we can't just use the simple permutation formula, which is for arranging distinct objects.
Instead, we use the formula: \[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \] This modified formula allows us to account for repeated items.

• \(n!\) represents the factorial of the total number of items.
• \(n_1!, n_2!, \ldots, n_k!\) are the factorials of frequencies of the identical items within the set.

Using this formula ensures that we only count unique permutations, avoiding overcounting due to repeated elements.

Multiset

A multiset is like a set, but with one key difference—elements can appear more than once. This concept is central to problems where distinguishable permutations are needed, as with letters or items that repeat.
For example, the sequence A, A, A, B, B, B, C, C, C has nine elements, but it's a multiset because the letters A, B, and C are repeated.
Analyzing multisets involves understanding both the total number of elements and their individual frequencies, which is crucial when applying the permutation formula with repeated elements, as it helps in determining how the repetitions affect possible permutations.

Factorials

Factorials play a critical role in permutations and combinations. The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\).
For instance, \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880\). When dealing with repeated elements, we use the factorial to manage these repetitions.

• For each group of repeated elements, we calculate their factorial. So for three A's, the factorial is \(3! = 6\).

These calculations help adjust the counting process to ensure each unique arrangement is counted just once, which is crucial for finding the number of distinguishable permutations.

Repeated Elements

Repeated elements occur when one or more items appear multiple times in a collection. In permutations, repeated elements can complicate counting unique arrangements, as they inherently limit the variety of orderings.
Consider a situation where we have the letters "A, A, A, B, B, B, C, C, C." If all letters were distinct, the permutations would simply be \(9!\). However, due to repetitions, many of these arrangements appear identical (e.g., swapping two "A" letters results in no visible change).
To accurately determine distinguishable permutations, the role of repeated elements is addressed through division in the permutation formula. By dividing by the factorial of each set of repeated elements \(n_1!, n_2!, \ldots\), we account for the indistinguishable swaps and thus only count valid unique permutations.