Problem 36 - Algebra and Trigonometry

Question

In how many ways can 5 different boxes be stacked?

Step-by-Step Solution

Answer

120

Step 1 - Understand the Problem

We need to determine the number of ways to stack 5 different boxes. This is a permutation problem since the order in which we stack the boxes matters.

Step 2 - Determine the Formula

For permutations of a set of distinct objects, the formula is given by the factorial of the number of objects. For 5 objects, this is represented as 5!.

Step 3 - Calculate the Factorial

Calculate the factorial of 5. Using the factorial formula: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]

Step 4 - Compute the Value

Multiply the numbers: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 5 - Conclude the Answer

The number of ways to stack 5 different boxes is 120.

Key Concepts

factorials

A factorial is a mathematical concept used frequently in permutations and combinatorics. It is represented by an exclamation mark \(!\). The factorial of a positive integer n, written as n!, is the product of all positive integers less than or equal to n.
For example:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Factorials grow very quickly as n increases. Factorials are essential for calculating permutations, which we'll discuss next.

combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and the counting of objects. In combinatorics, we often seek to answer how many ways we can arrange or select objects.
In the context of our problem with stacking 5 different boxes, we are looking for the number of permutations. A permutation considers the order of the objects, which means the arrangement matters.
The formula for the number of permutations of n distinct objects is n! (n factorial). This formula helps us determine how many unique ways we can arrange or order these objects.

distinct objects

A distinct object is one that is different or unique from others. When dealing with permutations or combinations, the nature of distinct objects is crucial because swapping any two changes the overall arrangement.
For instance, if we have 5 different boxes labeled A, B, C, D, and E, each box is distinct. This uniqueness means each box can occupy any position in the stack, creating a unique arrangement every time.
With 5 distinct boxes, we use the formula 5! (5 factorial) to determine the total number of ways we can stack them. Each permutation represents a unique order of these boxes. This is why we calculated 5! = 120 as the total number of distinct ways to stack the 5 different boxes.