Problem 24 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Letter Permutations How many permutations are possible from the letters of the word LOVE?

Step-by-Step Solution

Answer

24 permutations are possible.

Step 1: Analyze the Problem

Identify the type of problem we are dealing with. In this case, it is a permutations problem involving the arrangement of letters in a word.

Step 2: Count the Letters

Count the total number of distinct letters in the word "LOVE." There are four distinct letters: L, O, V, and E.

Step 3: Calculate the Permutations

Use the permutation formula for n distinct items, which is given by \( n! \) (n factorial). Here, \( n = 4 \). So we need to calculate \( 4! \).

Step 4: Compute the Factorial

Calculate \( 4! \) to find the number of permutations. \( 4! \) means \( 4 \times 3 \times 2 \times 1 = 24 \).

Step 5: Conclude the Solution

The total number of different permutations of the letters in the word "LOVE" is 24.

Key Concepts

Factorial

Factorials are a key mathematical concept used in counting and permutations. When you see the notation of a number followed by an exclamation mark, like "4!", it refers to a factorial. This expression means you multiply that number by every positive integer below it.
So, "4!" or 4 factorial is calculated as follows:

• Start with 4, then multiply by 3, which equals 12.
• Take that result (12) and multiply by 2, resulting in 24.
• Finally, multiply 24 by 1, which keeps it, of course, at 24.

For any number "n", the formula for a factorial is:\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \]Factorials are crucial in permutations and combinations because they help us calculate how many different ways we can arrange a set of objects. For the word "LOVE," understanding factorial makes calculating the total permutations straightforward!

Distinct Letters

When solving permutations involving words, it’s important to identify distinct letters. Distinct letters are the unique characters that appear in a word. For example, looking at the word "LOVE," we can see that each letter is different: L, O, V, and E.
Understanding distinct letters helps you to know what you're working with when calculating arrangements.

• Count each unique letter only once.
• Do not worry about repeated letters in this case, as they do not exist in "LOVE."

If a word had repeating letters, additional steps would be necessary to adjust for the repetitions, because repeated letters reduce the number of unique permutations you can make. But for "LOVE," it’s as simple as counting the four distinct letters you have.

Arrangement of Letters

The arrangement of letters is what permutations are all about. It refers to the different ways you can organize and order the letters in a word. For permutations, you arrange all items from a set, without excluding any. Each arrangement is a unique permutation.
When calculating permutations for the word "LOVE," you assess the complete orderings of its distinct letters. With four different letters, the permutations are calculated as "4!", resulting in 24 unique arrangements.
Consider each different sequence of the letters as an individual outcome, such as LOVE, EVOL, VLEO, and so on.

• All permutations use every letter once per arrangement.
• Permutations change with the inclusion or exclusion of letters.

Knowing how to arrange letters gives you insight into how flexible the letter order is and how many combinations there can be. Such understanding is foundational for grasping broader concepts in mathematics involving combinations and probability.