Problem 35 - Algebra and Trigonometry

Question

33–40 These problems involve distinguishable permutations. Arranging Coins In how many different ways can four pennies, three nickels, two dimes, and three quarters be arranged in a row?

Step-by-Step Solution

Answer

27,720 ways.

Step 1: Understand the Question

The problem asks for the number of ways to arrange a set of 12 coins, comprised of 4 pennies, 3 nickels, 2 dimes, and 3 quarters, in a row. This situation involves distinguishable permutations of objects.

Step 2: Identify Total Number of Coins

Identify the total number of coins to be arranged. Here, there are: 4 pennies, 3 nickels, 2 dimes, and 3 quarters. The total number is: \[ 4 + 3 + 2 + 3 = 12 \]

Step 3: Apply the Permutation Formula

Use the formula for permutations of multiset: \[ \frac{n!}{n_1! \times n_2! \times _3! \times n_4!} \]Where \( n \) is the total number of coins and \( n_1, n_2, n_3, \) and \( n_4 \) are the counts of each type of coin.

Step 4: Calculate Factorials

First, calculate the factorial of the total number of coins (12):\[ 12! = 479001600 \]

Step 5: Calculate Denominator Part 1

Calculate the factorial of the number of pennies, which is 4:\[ 4! = 24 \]

Step 6: Calculate Denominator Part 2

Calculate the factorial of the number of nickels, which is 3:\[ 3! = 6 \]

Step 7: Calculate Denominator Part 3

Calculate the factorial of the number of dimes, which is 2:\[ 2! = 2 \]

Step 8: Calculate Denominator Part 4

Calculate the factorial of the number of quarters, which is 3:\[ 3! = 6 \]

Step 9: Find the Distinguishable Permutations

Plug the values into the formula to find the number of distinguishable permutations:\[ \frac{12!}{4! \times 3! \times 2! \times 3!} = \frac{479001600}{24 \times 6 \times 2 \times 6} \]\[ = \frac{479001600}{1728} \]\[ = 27720 \]

Step 10: Conclusion

There are 27,720 different ways to arrange the four pennies, three nickels, two dimes, and three quarters in a row.

Key Concepts

Permutation Formula

Permutations are about arranging things in a specific order. The permutation formula is crucial when determining how many different ways we can organize items. In problems involving distinguishable permutations, the formula becomes slightly different because some items are identical. Here is where the multiset permutation formula comes into play, which is derived from dividing the factorial of the total number of items by the factorial of each group of identical items.

The formula looks like this:

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

where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) represent the number of identical items in each category. This method ensures we do not overcount permutations where the order among identical items doesn't matter. By understanding this formula, you can solve many permutation problems with ease.

Factorials

Factorials are a fundamental concept in combinatorics and permutations. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are used to calculate the total number of ways to arrange \( n \) distinct objects. In permutation problems, we often deal with the factorial of a number. The application of factorial is crucial, especially in solving multiset permutations, where factorials of identical items help adjust the overcounting in plain factorial calculations.

For instance, when calculating distinguishable permutations of coins, we compute the factorial for the total number of coins, then divide by the factorials of the group counts, which are like the 4 pennies, 3 nickels, and so forth. Mastering factorials will make solving such problems straightforward.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of arrangements. It encompasses a wide range of problems, but permutations are a key aspect. Distinguishable permutation problems, like arranging coins, fall under this area.

Combinatorics allows us to count without manually listing all possible arrangements. It uses principles like the permutation and combination formulas to solve problems efficiently. Combinatorial concepts enable us to analyze and calculate outcomes quickly when dealing with large sets of objects, as manually arranging them is often impractical.

By using combinatorial strategies, you can tackle complex tasks like organizing different coins. It empowers you to think about problems logically and find solutions systematically, which is an invaluable skill in mathematics and related fields.

Multiset Permutations

Multiset permutations involve arranging a set of objects, where some objects may repeat. Unlike regular permutations, where all items are distinct, multiset permutations must account for repeated items to avoid overcounting. We use a specific formula, derived from the basic permutation formula, which adjusts for these repetitions.

Consider a task like arranging coins with several duplicates, for example, four pennies, three nickels, two dimes, and three quarters. The multiset permutation approach helps us calculate how many unique ways we can do this. The formula is:

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

where \( n \) is the total number of items, and each \( n_i \) represents the count of identical objects in each subset. By using this formula, we accurately calculate permutations without redundancy, ensuring we accurately account for the repeated items' arrangements.