Problem 25 - Calculus for Biology and Medicine

Question

Four A's and five B's are to be arranged into a nine-letter word. How many different words can you form?

Step-by-Step Solution

Answer

126 different words can be formed.

Step 1: Understand the Problem

We need to form a nine-letter word using 4 A's and 5 B's. The problem is essentially asking us to count the number of distinct permutations of these letters.

Step 2: Calculate Total Permutations

If we were to arrange 9 distinct letters, there would be \(9!\) possible permutations, where \(!\) denotes the factorial of a number.

Step 3: Adjust for Repeated Letters

Since there are repeated letters, namely 4 A's and 5 B's, we must divide the total permutations by the factorial of the number of repeated letters to avoid overcounting: \(\frac{9!}{4! \times 5!}\).

Step 4: Compute Factorials

Calculate the necessary factorials: \(9! = 362880\), \(4! = 24\), and \(5! = 120\).

Step 5: Final Calculation

Plug the factorials into the formula: \(\frac{362880}{24 \times 120}\). This simplifies to \(\frac{362880}{2880}\).

Step 6: Calculate the Result

Perform the division: \(\frac{362880}{2880} = 126\). Thus, there are 126 different words that can be formed.

Key Concepts

Understanding Factorials

Factorials are essential in permutation and combinatorics problems. The factorial of a positive integer, denoted by the symbol \( ! \), is the product of all positive integers from 1 to that number. For example, the factorial of 4, written as \( 4! \), is equal to \( 4 \times 3 \times 2 \times 1 = 24 \). Factorials grow rapidly as the number increases; for instance, \( 9! = 362880 \).

They're crucial in counting arrangements since they give us the total number of ways to arrange a set number of distinct items. When dealing with repeated elements, though, we need to modify our initial count to account for these repetitions.

Exploring Distinct Permutations

Distinct permutations refer to the different ways of arranging a set of items where some items may be the same. When calculating permutations with repeated items, such as multiple letters in a word, we must ensure not to count the same arrangement more than once.

To adjust for repetitions, we divide the total permutations by the factorial of each set of identical items. For example, in a scenario with 4 A’s and 5 B’s, we would calculate the total as \( \frac{9!}{4! \times 5!} \). This formula prevents overcounting due to the indistinguishable nature of the repeated items.

• Calculate the overall permutations as if the items were all distinct: \( 9! \).
• Divide by the product of the factorials of the repeated items: \( 4! \) for A’s, \( 5! \) for B’s.

This ensures each unique arrangement is only counted once, giving us a total of 126 distinct permutations for the example with 4 A's and 5 B's.

The Role of Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns in sets. It plays a significant role in determining how many ways items can be chosen and arranged according to specific rules. When facing problems involving arrangements and combinations, combinatorics provides the tools needed to compute possibilities.

In our problem of arranging the letters in the word and their repetitions, combinatorics allows us to:

• Understand the concept of factorials for arranging distinct items.
• Use the distinct permutation formula to adjust for repeats.

By leveraging these concepts, we can solve problems efficiently without needing to physically arrange every possibility. This saves time and simplifies complex arrangements into manageable mathematical terms.