Problem 36 - Algebra 2 and Trigonometry

Question

In \(32-37,\) determine the number of different arrangements. Making a row of coins using 4 pennies, 3 nickels, and 3 dimes

Step-by-Step Solution

Answer

There are 4200 different arrangements of the coins.

Step 1: Recognize the Counting Problem

This problem is about arranging objects where some of them are identical. We need to find out how many distinct ways we can arrange these identical coins in a row.

Step 2: Identify Total Coins

First, count the total number of coins. We have 4 pennies, 3 nickels, and 3 dimes, which adds up to a total of 10 coins.

Step 3: Use the Formula for Arrangements with Repetition

The formula for arranging n objects where there are repetitions is given by the multinomial coefficient: \[ \frac{n!}{n_1! \cdot n_2! \cdot n_3! \ldots} \] In this case, \( n = 10 \). The repetitions are 4 pennies (\( n_1 = 4 \)), 3 nickels (\( n_2 = 3 \)), and 3 dimes (\( n_3 = 3 \)).

Step 4: Calculate Factorials

Calculate the factorials needed: \( 10! = 3628800 \), \( 4! = 24 \), and \( 3! = 6 \). These calculations can often be done using a calculator.

Step 5: Plug into the Formula

Substitute the factorial values into the formula: \[ \frac{10!}{4! \cdot 3! \cdot 3!} = \frac{3628800}{24 \cdot 6 \cdot 6} \]

Step 6: Simplify the Expression

Calculate the denominator: \( 24 \cdot 6 \cdot 6 = 864 \). Now perform the division: \( \frac{3628800}{864} = 4200 \).

Step 7: Conclude the Calculation

The number of different arrangements for the coins is 4200.

Key Concepts

Arrangements with Repetition

In our daily lives, we often come across situations where we need to arrange items, such as coins or letters, but some of these items might be indistinguishable from others. For instance, if you have a collection of 4 pennies, 3 nickels, and 3 dimes, you'll need to figure out how many unique ways you can lay them out in a line.

This is where the concept of **arrangements with repetition** becomes crucial. When some of the items are identical, it's important to use a specific formula that can account for these repetitions, ensuring each unique arrangement is counted correctly.

The formula for arranging a total of "n" items where some items are repeated is given by the multinomial coefficient:

• \[ \frac{n!}{n_1! \cdot n_2! \cdot n_3! \ldots} \]

Here, "n" represents the total number of objects you have. "n1", "n2", and so forth represent how many of each identical item you have. Like in our example, where you have 10 coins in total, consisting of 4 identical pennies, 3 identical nickels, and 3 identical dimes.

Using this formula allows for all possible positions or arrangements of these items to be explored and calculated, respecting the indistinguishable nature of similar items.

Factorials

A key part of calculating arrangements with repetition involves understanding factorials. But, what exactly is a factorial?

In mathematics, a factorial (denoted by the symbol "!") of a non-negative integer "n" is the product of all positive integers less than or equal to "n". For example, the factorial of 3, written as "3!", is the product 3 x 2 x 1, which equals 6.

Factorials grow very quickly as "n" increases, and they play an integral role in permutations and combinations problems. In our coin example, to find how many ways you can arrange pennies, nickels, and dimes, you'll calculate:

• The total number of coins factorial, "10!" (which gives us all possible arrangements without considering identical items),
• And the factorial of each set of identical coins: "4!" for pennies, and "3!" each for nickels and dimes.

It’s important to include these factorials for each type of item to accurately determine the number of distinct arrangements. Hence, in our example, these are calculated as follows:

• \( 10! = 3628800 \)
• \( 4! = 24 \)
• \( 3! = 6 \) for nickels and dimes.

These calculations, though sometimes requiring a calculator, are usually straightforward and provide essential information to solve the problem at hand.

Combinatorics

Combinatorics is a fascinating and broad area within mathematics that deals with counting, arranging, and combining objects. It answers how many ways you can choose or arrange a specific number of objects from a given set when order or repetition may or may not matter.

In situations like arranging coins, combinatorics provides structured methods and formulas to efficiently solve complex problems that involve large numbers of objects.

One of the common applications of combinatorics is solving permutation problems, where the sequence or order of arrangement is important. For our example, where we're arranging 10 coins that include repeats of pennies, nickels, and dimes, it helps determine the distinct arrangements possible through the use of permutations with repetition formula.

Combinatorics isn't just about numbers and formulas; it enhances problem-solving skills and enables the tackling of diverse scenarios ranging from simple arrangements to intricate puzzles in real-life situations.