Problem 1 - Algebra for College Students

Question

How many different arrangements of the letters A, B, C, D, E, and F can be made? 720

Step-by-Step Solution

Answer

There are 720 different arrangements.

Step 1: Understanding the Problem

We need to find the number of different ways to arrange the letters A, B, C, D, E, and F.

Step 2: Identify Total Elements

There are 6 different letters to arrange: A, B, C, D, E, and F.

Step 3: Apply Permutation Formula

Since we are arranging all the letters, use the formula for permutations of n distinct items: \[ n! \] where n is the total number of items.

Step 4: Calculate the Factorial

Calculate 6! (6 factorial), which is the product of all positive integers up to 6: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 5: Perform the Calculation

Carry out the multiplication: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

Step 6: Conclusion

The total number of different arrangements of the letters A, B, C, D, E, and F is 720.

Key Concepts

Factorial

When talking about permutations and arrangements, the concept of a factorial is fundamental. A factorial, denoted as \( n! \), is the product of all positive integers up to a given number \( n \). For example, \( 6! \) is shorthand for multiplying all whole numbers from 6 down to 1. So:

• Start with the number 6.
• Multiply it by every whole number below it until you reach 1.

In this exercise, \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]This equals 720, the number of ways to arrange six different letters. This concept is crucial in permutations because it allows us to count the total number of methods to organize items without repetition, particularly in sequences like this one. The importance of the factorial is its direct application in counting possibilities efficiently.

Arrangement

Arrangement in mathematics, particularly concerning permutations, refers to the specific order of items. When we calculate arrangements, we're dealing with sequences. For six unique letters, every different lineup counts as a distinct permutation.

• If you change the order of just one letter, it forms a new arrangement.
• For instance, swapping the positions of A and B among six letters provides a whole new setup.

Arrangements are a key aspect when determining permutations, and they depend completely on the order of items. Notably, the permutation formula used in this exercise \[ n! \]is entirely focused on these arrangements, assuming each element is placed thoughtfully in different sequences, contributing to the total count (in this case, 720). This emphasizes that arrangements occur only when the order of items matters.

Distinct Items

The essence of determining permutations of items largely relies on their distinct nature. In mathematics, distinct items mean each one is unique and identifiable, making them countable separately in a sequence.

• Distinct items cannot be interchanged without creating a new arrangement.
• With six distinct letters like A, B, C, D, E, and F, each letter holds a unique position in any given sequence.

This concept clarifies why we use the straightforward factorial formula \( n! \) for distinct items. Since no letters are repeated, calculating 6! encompasses all possible ways to distinguish these characters in different sequences. This ensures every potential sequence is counted once, providing an accurate total of arrangements.