Problem 8 - Thinking Mathematically

Question

Use the Fundamental Counting Principle to solve Exercises 1-12. You need to arrange ten of your favorite photographs on the mantel above a fireplace. How many ways can you arrange the photographs, assuming that the order of the pictures makes a difference to you?

Step-by-Step Solution

Answer

There are 3628800 ways to arrange the ten photographs on the mantel.

Step 1: Identify the variables

For this exercise, we know the total number of photographs (n) is 10. As we are arranging all of them, the number of photographs to choose (r) is also 10.

Step 2: Substitute into the permutation formula

Substituting into the permutations formula, we get 10P10 = 10! / (10-10)!.

Step 3: Calculate Factorial

Calculate 10! which equals \(10*9*8*7*6*5*4*3*2*1 = 3628800\). Because (10-10)! is 0!, and by definition, 0! equals 1, our equation simplifies to \(10P10 = 3628800 / 1\).

Step 4: Simplify the equation

Simplifying the equation leads to 10P10 = 3628800. Thus, there are 3628800 ways to arrange the ten photographs.

Key Concepts

permutations

When we talk about permutations, we are dealing with situations where the arrangement or order of items is important. This concept is crucial in many real-world scenarios where the position or sequence can change the outcome.
For example, in the context of arranging photographs on a mantel, each different sequence of photographs represents a unique permutation. If you change the order of just two photographs, it results in a completely different permutation. Therefore, permutations are all about specific arrangements.

The formula for finding permutations is represented as \( nP r = \frac{n!}{(n-r)!} \). Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items being chosen. In our mantel example, both \( n \) and \( r \) are 10 because all 10 photographs need arranging, meaning each arrangement matters.

factorial calculation

Factorials are fundamental in solving permutation problems. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).
For instance, calculating \( 10! \) (read as '10 factorial') involves multiplying all whole numbers from 1 to 10, which is \( 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800 \). This computation shows how factorials quickly grow into large numbers.

In permutation calculations, like arranging photographs, factorials help determine how many total possible arrangements exist. A special note is \( 0! \), which by definition equals 1, and is important when all items are chosen for an arrangement, simplifying the calculation of permutations.

arrangement problems

Arrangement problems revolve around finding how many ways we can sequence or order items. These problems often appear when the order of items has to be considered critically, such as when decorating a space or setting up a display.
Addressing an arrangement problem requires understanding if the order of items changes the outcome, which is a central characteristic of permutation-related problems.

In the exercise concerning photographing arrangement, identifying how many different ways you can arrange the photos means solving an arrangement problem. By applying the permutation formula and calculating the factorial, we find that the number of configurations for the ten photos is 3628800. This large number reflects all possible sequences in which the photos could be displayed, highlighting the importance of order in such tasks.