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Chapter 10: Problem 27

How many different photographs are possible if four children line up in a row?

Short Answer

Expert verified
The number of possible photographs is 24.

Step by step solution

01

Understanding the concept of permutation

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. In other words, permutation is the different arrangements that a set of elements can make. If the 'n' is the total number of elements and 'r' is the number of elements to select, then the permutation formula is given by P(n,r) = n! / (n-r)! where '!' denotes factorial.
02

Applying the permutation formula

In this task, there are 4 children, and all of them need to be arranged in a row. So, n=r=4. Now use the formula for Permutations: P(n, r) = n! / (n-r)! = 4! / (4-4)!.
03

Calculating the factorial

The factorial is calculated as 4! = 4 x 3 x 2 x 1 = 24 and (4-4)! is 0! We define 0! to be 1 (this is a convention in mathematics). Hence, 4! / (4-4)! = 24/1 = 24.

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Key Concepts

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

Factorial
The concept of a factorial is foundational in mathematics, especially in permutations and combinations. Put simply, the factorial of a number "n" (denoted as \( n! \)) is the product of all positive integers less than or equal to "n".
For example, the factorial of 4 is computed as follows:
  • Start with 4.
  • Multiply it by the next lower integer: 4 × 3.
  • Continue multiplying by the next lower integers: 4 × 3 × 2 × 1.
  • The result: \( 4! = 24 \).
It's important to remember that 0! is defined to be 1. This might seem puzzling, but it's a useful convention that helps formulas work out neatly in combinatorics and other mathematical fields.
Factorials grow very quickly, which is why they are so powerful in creating permutations.
Arrangement
Arrangement is a key idea when dealing with permutations. It refers to how objects can be ordered or sequenced. In permutations, the order of objects is crucial and changes the outcome.
Consider the four children needing to form a line. Each different sequence they can form is considered a unique arrangement. For example:
  • Order A-B-C-D is one arrangement.
  • Order B-A-C-D is another.
  • Thus, swapping positions results in a new arrangement.
In the given exercise, since we are arranging all four children, we don't leave any child out, which simplifies our calculation to just finding \( 4! \). This is typical for simple permutation problems where all elements of a set are used.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It's like mathematical organization and arrangement.
Permutations are part of combinatorics and specifically focus on the arrangement of items when order matters.
In contrast, combinations consider selection without regard to order.
Here鈥檚 a helpful mental picture of these concepts:
  • If you rearrange your books on a shelf, that鈥檚 a permutation.
  • If you simply choose which books to take on a trip and ignore the order, that鈥檚 a combination.
The problem you鈥檙e working through is a perfect example of applying combinatorial principles. It uses permutations to figure out how four children can arrange themselves in a line. Combinatorics plays a big part in probability, statistics, and everyday problem-solving.

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Most popular questions from this chapter

In this set of exercises, you will use sequences to study real-world problems. Joan is offered two jobs with differing salary structures. Job A has a starting salary of \(\$ 30,000\) with an increase of \(4 \%\) per year. Job B has a starting salary of \(\$ 35,000\) with an increase of \(\$ 500\) per year. During what years will Job A pay more? During what years will Job B pay more?

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