Problem 6 - Algebra and Trigonometry

Question

Use the formula for \(_{n} P_{r}\) to evaluate each expression. \(_{9} P_{9}\)

Step-by-Step Solution

Answer

\(_{9} P_{9} = 362,880\)

Step 1: Understanding the formula for Permutations

The formula for permutations is given by \(_{n} P_{r} = \frac{n!}{(n-r)!}\) where n is the total number of objects, r is the number of objects to choose and arrange, and ! denotes factorial. In our case, both n and r are 9, so the expression simplifies to \(_{9} P_{9} = \frac{9!}{(9-9)!}\).

Step 2: Simplify the denominator of the fraction

The denominator of the fraction is (9-9)!. This simplifies to 0!. According to the definition of factorial, 0! equals 1. So now the expression simplifies to \(_{9} P_{9} = \frac{9!}{1}\).

Step 3: Calculate the numerator of the fraction

The numerator, 9!, is the product of all positive integers from 1 to 9, i.e., 9! = 9*8*7*6*5*4*3*2*1 = 362,880. So \(_{9} P_{9} = \frac{362,880}{1}\).

Step 4: Final Evaluation

Since any number divided by 1 is the number itself, we find that \(_{9} P_{9} = 362,880\).

Key Concepts

Factorial Notation

Understanding factorial notation is crucial when delving into the world of permutations and combinations. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For instance, the factorial of 5, or 5!, is computed as the product of all positive integers from 1 to 5:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
This notation is essential because it lays the foundation for calculating how many different ways we can arrange or select items.

• The factorial of 0 is a special case and is defined to be 1, that is \( 0! = 1 \).
• As the numbers get larger, the value of the factorial grows exponentially, making it an important concept in evaluating combinatorial expressions.

Understanding factorial notation is not only about memorizing the process but also recognizing the pattern of the multiplying sequence, as this will help in grasping more complex mathematical concepts involving permutations and combinations.

Permutations and Combinations

In the realm of permutations and combinations, we explore the different ways in which a set of objects can be arranged or selected. These concepts are fundamental to probability and statistics, as well as to various real-world situations like scheduling events or encryption algorithms.

Permutations

Permutations refer to the arrangements of a certain number of objects taken from a larger set, where the order matters. For example, the number of ways to arrange three books on a shelf from a set of five is a permutation problem. The mathematical formula to calculate permutations is given by:
\( _{n}P_{r} = \frac{n!}{(n-r)!} \)
where \( n \) represents the total number of objects, and \( r \) represents the number of objects to choose and arrange.

Combinations

Conversely, combinations involve selecting items from a group where the order does not matter. The number of ways to pick three books from a shelf of five, regardless of how they are later arranged, is a question of combinations. The formula for combinations is:
\( _{n}C_{r} = \frac{n!}{r!(n-r)!} \)
Permutations and combinations are two sides of the same coin of arrangement and selection but serve distinct mathematical questions based on the importance of the order in the context.

Evaluating Expressions

Evaluating expressions is a critical skill in mathematics that involves simplifying or finding the value of expressions according to mathematical rules. When addressing permutations, as seen in the exercise \(_{9} P_{9}\), evaluation involves several steps including substitution, simplification, and arithmetic operations.
To evaluate permutations expressions effectively:

• Identify the values of \( n \) and \( r \) in the permutation formula.
• Apply the factorial notation to expand and calculate the necessary factorials.
• Substitute the computed factorials into the permutation formula.
• Simplify the expression by performing any divisions or multiplications.

In our example, we calculated \( 9! \) and recognized that \( 0! = 1 \), ultimately finding that \( _{9}P_{9} = 362,880 \). This step-by-step process is not only essential for arriving at the correct answer, but also it reinforces the understanding of the fundamental principles that underpin the permutations formula.