Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ P(5,2) $$

Step-by-Step Solution

Answer

Evaluating the given expression \(P(5,2)\), using the permutation formula, we find there are \(20\) different possible arrangements of 2 objects out of 5 total objects.

Step 1: Identify the values of n and r

In this case, n = 5 (total number of objects) and r = 2 (objects we want to arrange).

Step 2: Apply the permutation formula

Now we will apply the permutation formula to these values. \[P(5,2) = \frac{5!}{(5-2)!}\]

Step 3: Calculate the factorials

Next, calculate the factorial of n=5 and (n-r)=3. \[5! = 5\times4\times3\times2\times1 = 120\] \[3! = 3\times2\times1 = 6\] Now substitute these values back into the formula. \[P(5,2) = \frac{120}{6}\]

Step 4: Calculate the final result

Finally, divide 120 by 6 to get the number of permutations possible with n=5 and r=2. \[P(5,2) = 20\] Hence, there are 20 different possible arrangements of 2 objects out of 5 total objects.

Key Concepts

Factorial Notation

Understanding factorial notation is essential when diving into combinatorics and permutations. In mathematics, the factorial of a non-negative integer is the product of all positive integers less than or equal to that number. For example, the factorial of 4 is denoted as '4!' and is calculated as:
\(4! = 4 \times 3 \times 2 \times 1 = 24\).
It's important to also note that the factorial of 0 is defined as 1, that is \(0! = 1\).
This function grows very rapidly, for even numbers as small as 5 or 6, the result of the factorial can be in the hundreds or thousands. Factorial notation plays a pivotal role in different areas of mathematics but is particularly important in combinatorics. It is the foundation for calculating permutations and combinations, where we evaluate the number of ways in which objects can be arranged or selected.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as an art and as a science. It focuses on the principles of how objects can be combined, arranged, or chosen. Under this large umbrella, we encounter the concept of permutations, which is specific to the arrangement of objects in a particular order. This becomes especially useful for problems where order matters: think of it like choosing the batting lineup for a baseball team, where it significantly matters who bats first, second, and so on.
When faced with combinatorial problems, tools such as factorial notation become incredibly handy. Combinatorics is not just theoretical; it has practical implications in fields like computer science, physics, and even in our daily lives, where organizational tasks might require using these principles unconsciously.

Arrangement of Objects

The arrangement of objects, and in particular the number of possible arrangements, is a central question in the study of permutations. Permutations enable us to systematically calculate how many ways we can arrange a set of objects. The order of the elements is crucial, which differentiates a permutation from a combination, where the order is not considered.
Understanding the Exercise: In the example of \(P(5,2)\), which asks for the number of ways to arrange 2 objects from a set of 5, the different orders are what make each arrangement unique. This isn't like picking two candies from a bag, where we don't care which comes first; here, we do care about the order. We apply factorial notation to determine the number of arrangements – essentially, we compare the total amount of orderings for the full set to the leftover set when we remove the chosen objects (in the example, that’s the ordering of the remaining 3). Doing this lets us find the unique pairings without redundant counting.