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Chapter 14: Problem 28

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

Short Answer

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

Step by step solution

01

Identify the Total Number of Letters

The word 'LOVE' consists of 4 distinct letters: L, O, V, and E.
02

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 \( 4! \).
03

Calculate the Factorial

Calculate \( 4! \):\[4! = 4 \times 3 \times 2 \times 1 = 24\]
04

Interpret the Result

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Committee Membership A mathematics department consists of ten men and eight women. Six mathematics faculty members are to be selected at random for the curriculum committee. (a) What is the probability that two women and four men are selected? (b) What is the probability that two or fewer women are selected? (c) What is the probability that more than two women are selected?

An Identity Involving Combinations Kevin has ten different marbles, and he wants to give three of them to Luke and two to Mark. In how many ways can he choose to do this? There are two ways of analyzing this problem: He could first pick three for Luke and then two for Mark, or he could first pick two for Mark and then three for Luke. Explain how these two viewpoints show that $$ C(10,3) \cdot C(7,2)=C(10,2) \cdot C(8,3) $$ In general, explain why $$ C(n, r) \cdot C(n-r, k)=a(n, k) \cdot C(n-k, r) $$

Quality Control An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if any one of the ten is found to be defective, the process is halted and the machines are recalibrated. Suppose that at a certain time 5\(\%\) of the fuses being produced are actually defective. What is the probability that the assembly line is halted at that hour's quality check?

These problems involve combinations. Choosing Clothing If a woman has eight skirts, in how many ways can she choose five of these to take on a weekend trip?

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