Problem 28 - Algebra and Trigonometry

Question

These problems involve permutations. Letter Permutations How many permutations are possible from the letters of the word \(L O V E ?\)

Step-by-Step Solution

Answer

There are 24 permutations of the letters in 'LOVE'.

Step 1: Identify the Total Number of Letters

The word 'LOVE' consists of 4 distinct letters: L, O, V, and E.

Step 2: Apply Permutation Formula for Distinct Items

To find the number of permutations of 4 distinct letters, we use the factorial function. The formula for permutations of n distinct items is given by \( n! \). Here, \( n = 4 \), so we calculate \( 4! \).

Step 3: Calculate the Factorial

Calculate \( 4! \):\[4! = 4 \times 3 \times 2 \times 1 = 24\]

Step 4: Interpret the Result

There are 24 distinct permutations of the letters in the word 'LOVE'.

Key Concepts

Factorial

When it comes to permutations, one of the key mathematical concepts you must understand is the factorial. A factorial, denoted by the symbol '!', is a way of representing the product of all positive integers up to a specific number. It is a fundamental tool in determining permutations and combinations.
For example, when you see the notation \( n! \), it means:

• Multiply all whole numbers from the specified number down to 1.
• Use this result to compute permutations for distinct sets.

In the context of the word 'LOVE', which consists of four letters, we use the factorial \( 4! \) to find out how many different ways we can arrange the letters:\[4! = 4 \times 3 \times 2 \times 1 = 24\]
This calculation shows that there are 24 unique sequences we can create.

Distinct Letters

The concept of distinct letters is crucial when calculating permutations. A distinct letter means that each letter is unique and doesn't repeat. In the case of 'LOVE', all the letters L, O, V, and E are unique.

• Being distinct allows us to use the factorial to calculate permutations directly.
• If any letter repeated, we would need to adjust our calculations to account for those repetitions.

Knowing this helps ensure that the permutations calculated are truly different from each other, providing a set of unique sequences.

Combination Formula

While the exercise focuses on permutations, understanding combinations is also valuable. Combinations are different from permutations, as order does not matter in combinations. For permutations, order is crucial as it impacts the outcome.

• Combinations are calculated using a different formula: \( \frac{n!}{r! \times (n-r)!} \) where \( n \) is the total number of items, and \( r \) is the number of items to choose.
• In permutations, like the exercise on 'LOVE', order is important, hence we use \( n! \) directly.

This understanding provides more clarity when solving problems involving different arrangements of items.