Problem 33
Question
, \\#, \\#, @, \$, \%, \%, \%, \%, \%\( # For the following exercises, find the distinct number of arrangements. The symbols in the string \)\\#, \\#, \\#, @, \$, \%, \%, \%, \%, \%$
Step-by-Step Solution
Verified Answer
The distinct number of arrangements is 5040.
1Step 1: Identify Symbols and Frequencies
In the string given, there are several symbols: three '#' symbols, one '@' symbol, one '$' symbol, and five '%' symbols.
2Step 2: Determine Total Number of Symbols
Calculate the total number of symbols by summing the frequencies: \(3 + 1 + 1 + 5 = 10\).
3Step 3: Apply the Permutation Formula for Repeated Items
Use the permutation formula for arrangements with repetitions: \[\frac{n!}{n_1! \times n_2! \times n_3! \times \ldots}\] where \(n\) is the total number of symbols and \(n_1, n_2, \ldots\) are the frequencies of the repeated symbols.
4Step 4: Substitute Values into the Formula
Substitute the known values into the formula: \(n = 10\), \(n_1 = 3\) for '#', \(n_2 = 1\) for '@', \(n_3 = 1\) for '$', \(n_4 = 5\) for '%'. Calculate: \[\frac{10!}{3! \times 1! \times 1! \times 5!}\]
5Step 5: Calculate Factorials
Compute each factorial: \(10! = 3628800\), \(3! = 6\), \(5! = 120\). Using \(1! = 1\) simplifies our calculation.
6Step 6: Evaluate the Expression
Substitute the factorial values into the expression: \[\frac{3628800}{6\times1\times1\times120} = \frac{3628800}{720}\] Resulting in 5040.
Key Concepts
Factorial CalculationDistinct ArrangementsSymbols Frequency Count
Factorial Calculation
Factorial calculations are fundamental in permutation problems, especially when dealing with arrangements of different symbols. A factorial, denoted by an exclamation mark (!), is a product of all positive integers up to a given number. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
In our exercise, we need to calculate the factorial of 10, which is the total number of symbols involved. Therefore, \(10!\) is calculated as \(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800\).
Another critical aspect is computing smaller factorials for the repeated symbols like \(3!\) for `#`, which is \(3 \times 2 \times 1 = 6\), and \(5!\) for `%`, which is already covered as 120.
Understanding how to calculate these factorials is essential because they help in determining the number of distinct arrangements by adjusting repetitions.
In our exercise, we need to calculate the factorial of 10, which is the total number of symbols involved. Therefore, \(10!\) is calculated as \(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800\).
Another critical aspect is computing smaller factorials for the repeated symbols like \(3!\) for `#`, which is \(3 \times 2 \times 1 = 6\), and \(5!\) for `%`, which is already covered as 120.
Understanding how to calculate these factorials is essential because they help in determining the number of distinct arrangements by adjusting repetitions.
Distinct Arrangements
Distinct arrangements refer to the total number of unique ways that symbols can be laid out while respecting their order and frequency. This concept becomes particularly interesting when some symbols repeat, requiring adjustments in the permutation calculation.
In situations where symbols repeat, we use the permutation formula for repeated items: \[\frac{n!}{n_1! \times n_2! \times n_3! \times \ldots}\]where \(n!\) indicates the factorial of the total number of items, while \(n_1!\), \(n_2!\), and so on refer to the factorials of each group's frequency of repetition.
In our example, the symbols '#', '@', '$', and '%' repeat with frequencies of 3, 1, 1, and 5, respectively, and the total symbols count to 10. Using the formula, we calculated the distinct arrangements as:\[\frac{10!}{3! \times 1! \times 1! \times 5!} = \frac{3628800}{720} = 5040\]
This calculated number tells us there are 5040 unique ways to arrange these symbols.
In situations where symbols repeat, we use the permutation formula for repeated items: \[\frac{n!}{n_1! \times n_2! \times n_3! \times \ldots}\]where \(n!\) indicates the factorial of the total number of items, while \(n_1!\), \(n_2!\), and so on refer to the factorials of each group's frequency of repetition.
In our example, the symbols '#', '@', '$', and '%' repeat with frequencies of 3, 1, 1, and 5, respectively, and the total symbols count to 10. Using the formula, we calculated the distinct arrangements as:\[\frac{10!}{3! \times 1! \times 1! \times 5!} = \frac{3628800}{720} = 5040\]
This calculated number tells us there are 5040 unique ways to arrange these symbols.
Symbols Frequency Count
Knowing the frequency count of each symbol is crucial to employing the permutation formula correctly. This concept involves counting how many times each symbol appears in the set.
In our version of the puzzle, we have:
This careful counting ensures we accurately reflect the real distinctiveness of each arrangement possible, leading to a proper computation of unique permutations as seen with this exercise.
In our version of the puzzle, we have:
- `#` appears 3 times.
- `@` appears 1 time.
- `$` appears 1 time.
- `%` appears 5 times.
This careful counting ensures we accurately reflect the real distinctiveness of each arrangement possible, leading to a proper computation of unique permutations as seen with this exercise.
Other exercises in this chapter
Problem 33
For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{8.9,10.3,11.7, \ldots\\} $$
View solution Problem 33
For the following exercises, write the first eight terms of the sequence. $$ a_{1}=2, a_{2}=10, a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}} $$
View solution Problem 33
Write the first eight terms of the sequence. $$a_{1}=2, a_{2}=10, a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}$$
View solution Problem 34
For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum of 3.
View solution