Problem 23 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Three-Letter Words How many three-letter “words” can be made from the letters FGHIJK? (Letters may not be repeated.)

Step-by-Step Solution

Answer

120 three-letter words can be made.

Step 1: Determine the Total Number of Letters

There are 6 letters available: F, G, H, I, J, K.

Step 2: Choose the First Letter

For the first position, we have a total of 6 choices since we can choose any of the letters.

Step 3: Choose the Second Letter

Once the first letter has been chosen, we cannot repeat it. Therefore, for the second position, only 5 choices remain.

Step 4: Choose the Third Letter

After choosing the first two letters, we have used up two of our letters. Thus, for the third position, only 4 choices are left.

Step 5: Calculate the Number of Permutations

The total number of three-letter "words" (permutations) is the product of the numbers of choices for each position: \( 6 \times 5 \times 4 \).

Step 6: Perform the Multiplication

Calculate the expression: \( 6 \times 5 \times 4 = 120 \).

Key Concepts

Counting Problems

Counting problems are a fascinating part of mathematics that help us determine the number of possible arrangements in a given situation. In the case of the three-letter words made from the letters F, G, H, I, J, and K, we are dealing with a specific type of counting problem called permutations. Permutations are used when the order of items matters, like in the formation of these words.

Here, the task is to calculate how many different arrangements can be made with three letters, chosen from a set of six. Each letter can be used only once in every word, which means repetition is not allowed.

To solve this, we follow a sequential selection approach:

• First, decide on the first letter. We have 6 options available.
• Then, pick the second letter, but since the first letter cannot be used again, we have only 5 options.
• Lastly, choose the third letter from the remaining 4 options.

Combining these steps gives us the total number of possible permutations, or arrangements, for the problem.

Combinatorics

Combinatorics is an essential area of mathematics focused on counting, arrangement, and combinations of objects. It's a toolset that allows us to handle and solve counting problems efficiently. Permutations, part of combinatorics, are specifically useful when the sequence of items is important.

In our three-letter word example, we need to understand how different sequences are formed. Since we start with 6 different letters, we progressively reduce the number of choices with each selection:

• First, from 6 letters, we pick one.
• Next, from the remaining 5, select the second letter.
• Finally, choose the third letter from 4 left.

The number of permutations is calculated as the product of options available at each step. This systematic approach is a core concept in combinatorics, helping to break down complex problems into manageable steps.

Combinatorics allows us to extrapolate these principles to more extensive and diverse scenarios, ensuring that no possible arrangement is overlooked.

Factorial Computation

Factorial computation is a fundamental mathematical concept often used in permutation calculations. The factorial of a number is the product of all positive integers less than or equal to that number. It's denoted by an exclamation mark, for example, factorial of 6 is written as \(6!\).

In problems involving permutations, factorials make calculations straightforward, especially when dealing with large sets of objects. However, in the case of partial permutations (like our three-letter word problem), it's a little more involved.

For this exercise, although we don't use full factorials directly, understanding that \( n! \) represents the total permutations of \( n \) distinct items helps us comprehend the sequential element choices:

• We start with 6, then move to 5, and end with 4 choices.
• This sequence is similar to a partial factorial, \(6 \times 5 \times 4\), reflecting the calculations needed for creating the three-word combinations.

Grasping the factorial concept deepens our understanding of the logic behind permutation results, illustrating how different arrangements are inherently calculated through systematic reduction and multiplication.