Problem 37 - Discrete Mathematics with Applications

Question

Using the recursive definition of \(P(n, r),\) evaluate each. $$P(5,4)$$

Step-by-Step Solution

Answer

After evaluating each permutation using the recursive definition, we find that \(P(5,4) = 120\).

Step 1: Apply the recursive definition

We'll start by using the recursive definition of the permutation: $$ P(n, r) = n \times P(n - 1, r - 1) $$ For the given exercise, we need to find $$P(5, 4)$$. By applying the recursive definition, we get: $$ P(5, 4) = 5 \times P(4, 3) $$

Step 2: Calculate P(4, 3)

Now, we need to find $$P(4, 3)$$. Applying the recursive definition again: $$ P(4, 3) = 4 \times P(3, 2) $$

Step 3: Calculate P(3, 2)

We need to find $$P(3, 2)$$ next. Applying the recursive definition again: $$ P(3, 2) = 3 \times P(2, 1) $$

Step 4: Calculate P(2, 1)

Now we have to find $$P(2, 1)$$. Applying the recursive definition one last time: $$ P(2, 1) = 2 \times P(1, 0) $$

Step 5: Calculate P(1, 0)

When $$r = 0$$, the permutation is simply $$1$$ because there's only one way to arrange no elements. So, $$P(1, 0) = 1$$.

Step 6: Substitute the results back

Now that we've calculated each permutation, we can substitute the results back into the previous steps. \(\begin{aligned} P(2, 1) &= 2 \times P(1, 0) \\ &= 2 \times 1 \\ &= 2 \end{aligned}\) \(\begin{aligned} P(3, 2) &= 3 \times P(2, 1) \\ &= 3 \times 2 \\ &= 6 \end{aligned}\) \(\begin{aligned} P(4, 3) &= 4 \times P(3, 2) \\ &= 4 \times 6 \\ &= 24 \end{aligned}\) \(\begin{aligned} P(5, 4) &= 5 \times P(4, 3) \\ &= 5 \times 24 \\ &= 120 \end{aligned}\)

Step 7: Final Answer

After evaluating each permutation using the recursive definition, we find that $$P(5, 4) = 120$$.

Key Concepts

Permutation Calculation

Permutation calculation is all about arranging or ordering elements. Let's say you have a set of items and you want to know how many different ways you can arrange a subset of them.
This is where permutations come in. They are particularly useful in solving problems related to ordering.

To calculate permutations, we use the formula for a permutation of selecting \(r\) elements from \(n\) elements, denoted as \(P(n, r)\).
The basic permutation formula is:

• \(P(n, r) = \frac{n!}{(n-r)!}\)

This formula tells us that for a set of \(n\) elements, we take \(r\) elements at a time, arranging each combination uniquely.

In the given problem, applying this directly might seem difficult. That's why the recursive method is so handy.

Recursive Definition

A recursive definition is a way of defining something in terms of a smaller instance of itself.
This is a very natural way to solve problems that can be broken down into smaller components.

In the context of permutation calculations, the recursive definition allows us to compute \(P(n, r)\) in steps, rather than all at once.
This recursive approach utilizes the relationship:

• \(P(n, r) = n\times P(n-1, r-1)\)

This means that to find \(P(n, r)\), we just need to multiply \(n\) by the permutation of one less element taken and one less being chosen.

The process repeats until we reach a simple base case, like \(P(n, 0) = 1\), as choosing none results in just one option: doing nothing.

Discrete Mathematics

Discrete mathematics is an essential branch of mathematics, dealing with countable, distinct elements. Unlike calculus, which is about continuous change, discrete mathematics focuses on things that can be counted;
like objects, configurations, and sets.

This field is critical for computer science as it lays the mathematical foundations for understanding algorithms, data structures, and programming.
Permutation problems, like the one in the exercise, are a key area of study within discrete mathematics, as they involve arranging elements in particular orders.

By learning how to effectively use recursive definitions and permutation calculations, students gain valuable tools and insights. These skills apply to solving complex real-world problems, such as scheduling or data organization.