Problem 5 - College Algebra

Question

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{6} P_{6} $$

Step-by-Step Solution

Answer

The value of \(_6P_6\) is 720.

Step 1: Understand the Permutation Formula

The formula for permutation \(_nP_r\) is \(n! / (n - r)!\) . Permutations are the different ways in which we can arrange r objects from n distinct objects. The \('!'\) symbol denotes factorial, which is the product of all positive integers upto that number.

Step 2: Substitute the values into the formula

In this case, to find \(_6P_6\), we substitute n = 6 and r = 6. Notice that since n = r, the second part of the formula \((n - r)! = (6 - 6)! = 0!\) becomes 1, as \(0! = 1\). Thus, \(_6P_6\) essentially reduces to 6!.

Step 3: Calculate the Factorial

The next step is to calculate 6!, which is \(6 * 5 * 4 * 3 * 2 * 1 = 720.\)

Key Concepts

Permutation formula

Permutations are a fundamental concept in mathematics, specifically in combinatorics. Permutation refers to the arrangement of objects in a specific order. The permutation formula, denoted as \(_nP_r\), helps in determining the total number of ways in which \(r\) objects can be selected from \(n\) distinct objects and arranged in order.
The formula is given by:\[_nP_r = \frac{n!}{(n-r)!}.\]

• \(n\) is the total number of objects available.
• \(r\) is the number of objects to select and arrange.
• The exclamation mark \(!\) represents a factorial, which is discussed in detail below.

A key thing to remember about permutations is that the order of arrangement matters. This distinguishes permutations from combinations, where the order does not matter.

Factorial

Factorials are a simple yet crucial concept when working with permutations and combinatorics. The factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\).
Mathematically, it is expressed as:

• \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).

Factorials grow very quickly with higher numbers. For instance, calculating \(6!\) results in:

• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Interestingly, \(0!\) is defined to be \(1\). This definition is important for maintaining the consistency of mathematical formulas, particularly in permutations and combinations.

Combinatorics

Combinatorics is a branch of mathematics primarily concerned with counting, arrangement, and combination of sets of elements. It forms the backbone for concepts like permutations and combinations.
In combinatorics:

• Permutations are utilized when the order of arrangement of elements is crucial.
• Combinations are used when the order does not matter.

Understanding combinatorics is essential as it applies to many real-world problems, such as scheduling, seating arrangements, and game theory. By mastering the basic formulas and principles, students can solve complex counting and arrangement problems efficiently.