Problem 26 - Algebra for College Students

Question

A supply boat must stop at 9 oil rigs in the Gulf of Mexico. How many different routes are possible?

Step-by-Step Solution

Answer

362880 different routes are possible.

Step 1: Understand the Problem

Determine the total number of oil rigs the supply boat must visit and recognize this as a permutation problem since the order in which the supply boat visits the oil rigs matters.

Step 2: Set Up the Permutation Formula

Recall the formula for permutations of a set of items: \[ P(n) = n! \]where \(n!\) denotes the factorial of \(n\). For this problem, set \(n = 9\).

Step 3: Calculate the Factorial

Calculate \(9!\): \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 4: Perform the Multiplication

Perform the multiplication step by step: \(9! = 9 \times 8 = 72\), \(72 \times 7 = 504\), \(504 \times 6 = 3024\), \(3024 \times 5 = 15120\), \(15120 \times 4 = 60480\), \(60480 \times 3 = 181440\), \(181440 \times 2 = 362880\), \(362880 \times 1 = 362880\).

Step 5: State the Final Answer

The total number of different routes possible for the supply boat to visit the 9 oil rigs is 362880.

Key Concepts

Factorial Calculation

In permutation problems, understanding how to calculate a factorial is vital. A factorial, represented by \( n! \), is the product of all positive integers up to a given number \( n \).

For example, \( 9! \) means multiplying all the numbers from 9 down to 1:
\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]

Factorials grow very quickly with increasing numbers. Knowing how to calculate them accurately is crucial for solving permutation problems. Try practicing with smaller numbers to build your confidence and accuracy!

Order of Operations in Permutations

In a permutation problem, the order in which items are arranged or selected is paramount. For example, if you are arranging 9 oil rigs in different orders, each order is unique and counts as a separate permutation.

This differentiates permutations from combinations, where the order does not matter. Always identify if the problem specifies that order matters - this will tell you that you're dealing with a permutation problem.

Here's an example to illustrate this:

• If there are 3 items (A, B, and C), the permutations would include ABC, ACB, BAC, BCA, CAB, and CBA.
• As you can see, changing the order of items results in a different selection, making it a permutation problem.

Permutation Formula

The permutation formula helps calculate the total number of ways to arrange \ n \ items. The formula is:
\[ P(n) = n! \]

In this case, calculating the permutations of 9 oil rigs would be:

• Step 1: Determine the value of \ n \ (which is 9).
• Step 2: Substitute it into the permutation formula:
  \[ P(9) = 9! \]
• Step 3: Perform the factorial calculation:
  \[ 9! = 362880 \]

Thus, there are 362880 possible routes for the supply boat to take to visit 9 oil rigs. Understanding this formula and how to apply it will make solving permutation problems straightforward and manageable.