Problem 14 - College Algebra with Modeling and Visualization

Question

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. No \(a^{\prime \prime}\) s, three \(b^{\prime}\) s

Step-by-Step Solution

Answer

Only 1 distinguishable string can be formed.

Step 1: Understand the problem

We need to determine how many unique strings can be formed using a specific set of letters, in this case, no 'a's and three 'b's. Since there are no 'a's, all the characters in the string will be 'b's.

Step 2: Identify the possible permutations

Given that the string consists of only 'b's, we need to calculate the number of unique permutations of these 'b's. Since all characters are identical, there is only one unique way to arrange them.

Step 3: Calculate the number of permutations

The formula to calculate permutations when elements are identical is: \( \frac{n!}{p_1! \cdot p_2! \cdots} \). Here, \( n \) is the total number of characters, and \( p_1, p_2, \ldots \) are the counts of identical characters. Since there are only three 'b's, the formula becomes \( \frac{3!}{3!} = 1 \).

Key Concepts

Distinguishable Strings

When creating strings from a set of letters, some strings can be considered distinguishable, while others may not be. Distinguishable strings occur when the arrangement of letters creates unique string sequences.
Consider a simple case where you have different letters like 'a' and 'b'. Combining these differently forms strings like 'ab' and 'ba'.
Each of these arrangements is unique in its order, making them distinguishable.

• If all characters in the string are identical, such as all 'b's in our example, there is only one way to arrange them.
• This results in just one single distinguishable string.

In cases where there are no distinct letters, like having all 'b's, the concept of distinguishable strings is straightforward because you don't have different letters creating unique patterns.

Factorials

Factorials are a mathematical operation used to determine the number of ways to arrange a set of items. Represented by an exclamation mark (!), the factorial of a number, such as 3, is the product of all positive integers up to that number, e.g., \[3! = 3 \times 2 \times 1 = 6\]Factorials are often used in permutation calculations to determine different arrangements.
In permutation problems, such as finding unique strings, factorials help describe how many total possible arrangements exist.

• If characters are all different, the factorial of the number of characters tells you how many different layouts you can make.
• If characters are identical, factorials are used in both the numerator and denominator of the formula to adjust for those repeated items.

Factorials simplify the calculation process by offering a concise method to compute arrangements in both simple and complex problems.

Identical Items

Identical items in permutations involve items that are not distinct from each other.
For example, if you have three 'b's, as stated in our problem, it's a case of identical items. The permutation formula for identical items adjusts the number of possible arrangements by dividing the total permutations by the factorial of the counts of identical items.

• The formula for the permutations of identical items is:\[ \frac{n!}{p_1! \cdot p_2! \cdots} \]where \(n\) is the total number of items, and \(p_1, p_2, \ldots\) are the numbers of identical items.

When all items are identical, as with the three 'b's, every arrangement looks the same.
Therefore, even though the mathematical process takes place, it results in only one distinct outcome, as shown in the original solution: \[ \frac{3!}{3!} = 1 \]Understanding identical items aids in recognizing when and why permutations reduce to fewer unique outcomes.