Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ P(6,6) $$

Step-by-Step Solution

Answer

The expression \(P(6,6)\) represents the number of ways to arrange 6 items, taking all 6 at a time. Using the formula \(P(n, k) = \frac{n!}{(n-k)!}\), we find that \(P(6, 6) = \frac{720}{1} = 720\). Therefore, there are 720 different ways to arrange 6 items, taking all 6 at a time.

Step 1: Identify n and k

In this problem, we are given n = 6 and k = 6.

Step 2: Calculate n! and (n-k)!

Now, we need to calculate 6! and (6-6)!: \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\] \[(6-6)! = 0! = 1\]

Step 3: Calculate P(n, k) using the formula

\[P(6, 6) = \frac{6!}{(6-6)!}\] \[P(6, 6) = \frac{720}{1} = 720\] Therefore, there are 720 different ways to arrange 6 items, taking all 6 at a time.

Key Concepts

Factorial Notation

Factorial notation is fundamental to understanding permutations and combinations in mathematics. It’s represented by an exclamation mark (!) and involves multiplying a set of decreasing natural numbers down to one. When you see an expression like \( n! \), it refers to the product of all positive integers from \( n \) down to 1.

For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It's important to remember that \( 0! \) is defined to be 1, which often initially confuses students. This peculiar definition simplifies many mathematical formulae. In permutations, factorial notation is used to calculate the total number of different ways to arrange a given set of items without repetitions.

Counting Principle

The counting principle, also known as the fundamental counting principle, is used to calculate the total number of outcomes when there are multiple stages or events. It states that if one event can occur in \( m \) ways and another independent event can occur in \( n \) ways, then the total number of combined outcomes for the two events is \( m \times n \).

This principle can be extended to any number of events. It's particularly handy when dealing with permutations, where we want to know the number of possible arrangements of a set. If we’re arranging \( n \) items, and there are \( n \) options for the first item, \( n-1 \) for the second, and so on, the counting principle tells us that the total number of arrangements is the product of all these numbers, which leads us into factorial notation and computations.

Arrangements in Probability

In probability, we often deal with the concept of arrangements, particularly when we calculate the likelihood of certain orders or selections. This typically involves permutations, which assess how many ways we can arrange a subset of items from a larger pool. The general formula for permutations is given by \[ P(n, k) = \frac{n!}{(n-k)!} \], where \( n \) is the total number of items and \( k \) is the number of items being arranged.

Using our exercise as an example, \( P(6,6) \) calculates the number of ways to arrange 6 items out of 6, which logically implies using all the items. This is where the factorial function makes an appearance, simplifying the process to calculating just \( 6! \). Understanding this helps in dealing with more complex arrangements, especially where only a portion of the set is used, thereby broadening one's grasp of probability scenarios.