Problem 15 - Discrete Mathematics with Applications

Question

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$P(n, r)=P(n, n-r)$$

Step-by-Step Solution

Answer

The given statement \(P(n, r) = P(n, n-r)\) is true for nonnegative integers 'n' and '0 ≤ r ≤ n', as the comparison between the left-hand side and the right-hand side of the equation demonstrates that both sides are equal: \(\frac{n!}{(n-r)!} = \frac{n!}{r!}\).

Step 1: Understand the problem

We are given the statement: $$ P(n, r) = P(n, n-r). $$ Our task is to determine if this statement is true or false for nonnegative integers n and r, and given that \(0 \leq r \leq n\).

Step 2: Recall the formula for permutations

Permutations can be defined by the formula: $$ P(n, r) = \frac{n!}{(n-r)!}, $$ where the exclamation mark '!' denotes a factorial. The factorial of a non-negative integer 'n' is the product of all positive integers less than or equal to 'n'.

Step 3: Check the truth of the given statement

Now we need to check if: $$ \frac{n!}{(n-r)!} = \frac{n!}{(n-(n -r))!}. $$ The right-hand side simplifies to: $$ \frac{n!}{(n-(n -r))!} = \frac{n!}{r!}. $$ Comparing the left-hand side and the right-hand side, we can see that both sides are equal, as: $$ \frac{n!}{(n-r)!} = \frac{n!}{r!}. $$ Hence the given statement: $$ P(n, r) = P(n, n-r) $$ is true for nonnegative integers 'n' and '0 ≤ r ≤ n'.

Key Concepts

Factorial Notation

Factorial notation is a crucial concept when dealing with permutations. It's represented by the symbol '!', which stands for a factorial. In mathematics, a factorial of a non-negative integer \( n \) is the product of all positive integers from 1 to \( n \). For example, the factorial of 3, written as \( 3! \), is calculated as \( 3 \times 2 \times 1 = 6 \). Factorials are foundational in various areas like probability and statistics because they help calculate permutations and combinations. They help determine the number of ways to organize or arrange a set of items.
To provide a quick overview:

• \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \)
• The factorial of 0 is defined as 1: \( 0! = 1 \)
• Factorials grow very fast, much faster than exponential functions

Factorials are especially useful in permutations, where the order of arrangements matters.

Nonnegative Integers

A nonnegative integer is a number belonging to the set that includes all positive integers and zero. In other words, nonnegative integers are whole numbers that are zero or greater. This category excludes negative numbers, fractions, and decimals.
When discussing mathematics, nonnegative integers are essential for many calculations because they ensure that results are logical and consistent in certain mathematical theories and practical applications.
Here's why nonnegative integers matter:

• Permutations often require nonnegative integers for calculations, as seen in the permutations formula \( P(n, r) \).
• They provide boundaries for mathematical operations, simplifying problem-solving approaches.
• They play a vital role in sequences, series, and number theory.

Understanding nonnegative integers helps in effectively solving problems related to permutations and similar concepts.

Mathematical Proof

In mathematics, a proof is a logical argument demonstrating the truth of a given statement. It involves a series of deductions from known facts, axioms, and previously established statements. Proofs are central to validating mathematical concepts and ensuring that the assertions made are indeed true.
To appreciate the nature of a proof in permutations, consider the expression \( P(n, r) = P(n, n-r) \). This problem is an exercise in proving equality through algebraic manipulation and substitution.

Steps generally involved in a proof include:

• Clearly stating the problem.
• Identifying known information and relevant formulas (like the formula for permutations \( P(n, r) = \frac{n!}{(n-r)!} \)).
• Applying logical reasoning and simplification to verify the statement.
• Arriving at a conclusion that is consistent with the initial assumptions.

Using a proof provides the certainty needed to affirm that \( P(n, r) = P(n, n-r) \) is indeed true for all nonnegative integers \( n \) and \( 0 \leq r \leq n \). This approach solidifies understanding and adopts rigor, ensuring mathematical credibility.