Problem 17 - Algebra 2

Question

Evaluate each expression. \(_{5} P_{3}\)

Step-by-Step Solution

Answer

60

Step 1: Understand the permutation formula

To evaluate \( _{n}P_{k} \), understand that this represents the number of permutations of n items taken k at a time. The formula for finding permutations is \( _{n}P_{k} = \frac{n!}{(n-k)!} \) where \( n! \) means 'n factorial', the product of all positive integers from 1 to n.

Step 2: Apply the permutation formula to \( _{5}P_{3} \)

Now substitute n with 5 and k with 3 into the permutation formula. \( _{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5!}{2!} \)

Step 3: Calculate factorials

Calculate the factorials of the numbers. We know that \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) and \(2! = 2 \times 1 = 2\).

Step 4: Complete the calculation

Now divide the factorial of 5 by the factorial of 2 to get the number of permutations: \( \frac{5!}{2!} = \frac{120}{2} = 60 \) Thus, \( _{5}P_{3} = 60 \)

Key Concepts

Factorial Notation

Factorial notation is a fundamental concept in mathematics, specifically in the field of combinatorics. It is represented by an exclamation mark (!) following a number and is called 'factorial'. It stands for the product of all positive integers up to that number. For example, the factorial of 4, written as \(4!\), is calculated by multiplying all positive whole numbers from 1 to 4:
\(4! = 4 \times 3 \times 2 \times 1 = 24\).

In the context of permutations and combinations, factorial notation helps to count the different ways in which objects can be arranged or selected. A special case is \(0!\), which is defined to be 1. This might seem odd at first, but it's essential for the mathematical definition of permutations and combinations to work correctly, even when no objects are chosen or arranged.

Importance in Permutations

When calculating permutations, factorial notation is used to determine the total number of arrangements. For instance, in our exercise to find \(_{5}P_{3}\), we used the concept of factorial to simplify the expression. Factoring in the usefulness of factorial notation, it is also essential to mention that factorials grow very fast with larger numbers, which is why the concept becomes very significant when dealing with large sets of objects.

Combinatorics

Combinatorics is a branch of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, including algebra, probability, and geometry, and has applications in many fields such as computer science, physics, and statistics.

Within combinatorics, there are several subfields, including graph theory, enumeration, and design theory. However, at a basic level, combinatorics deals with the principles of counting and arranging objects according to specified rules. This can be as simple as figuring out how many different ways there are to arrange books on a shelf, or as complex as the many different structures of a graph or network.

The Role in Permutations and Combinations

In regards to permutations and combinations, which are both subsets of combinatorics, this discipline offers the theoretical foundation for understanding how different configurations are possible. In our exercise, we delve into the combinatorics of permutations, examining the specific instance of arranging 5 items 3 at a time.

Permutations and Combinations

Permutations and combinations are two concepts in combinatorics that deal with the ways in which objects can be selected and arranged from a larger set. Both play a critical role in probability and decision-making.

Permutations focus on the arrangement of objects where the order of arrangement is important. For instance, the sequence of letters ABC is different from BCA, and this distinction is crucial in permutations. The general formula for permutations is given by \(_{n}P_{k} = \frac{n!}{(n-k)!}\), where \(n\) is the total number of objects, and \(k\) is the number of objects to be arranged.

Combinations, on the other hand, deal with the selection of objects where the order doesn't matter. For example, choosing two fruits, an apple and a banana, out of a basket containing five varieties is the same as selecting a banana and an apple; the order of selection is irrelevant. The formula for combinations is \(_{n}C_{k} = \frac{n!}{k!(n-k)!}\).

Understanding Both Concepts

Understanding the difference between permutations and combinations is critical in solving many probability and counting problems. While permutations are about arrangements with regard to order, combinations are about selections without regard to order. Together, these concepts allow for a structured approach to solving various complex problems that involve counting arrangements and selections, as in the exercise where we calculate permutations for selecting 3 items from a set of 5.