Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{5} P_{5}$$

Step-by-Step Solution

Answer

The value of \(_{5} P_{5}\) is 120.

Step 1: Identify n and r

In this problem, \(_{5} P_{5}\), n and r are both 5.

Step 2: Calculate n!

Calculate the factorial of 5, which is \(5! = 5 * 4 * 3 * 2 * 1 = 120\).

Step 3: Calculate (n - r)!

Since n - r = 5 - 5 = 0, and the factorial of 0 (\(0!\)) is defined as 1.

Step 4: Substitute into the formula

Substitute the calculated n! and (n - r)! in the formula \(_{n} P_{r} = \frac{n!}{(n - r)!}\). Which gives, \(_{5} P_{5} = \frac{5!}{0!} = \frac{120}{1} = 120.\)

Key Concepts

Factorials

Factorials are fundamental to understanding permutations and combinations. A factorial, represented by an exclamation mark "!", indicates that we multiply a series of descending natural numbers. For example, the factorial of 5, denoted as \(5!\), is the product \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are essential because they help in calculating permutations and combinations, showing how many different ways you can arrange or choose items. A notable exception is \(0!\), which equals 1. This is because there is exactly one way to arrange nothing at all. This special definition keeps the calculations consistent in formulas involving factorials.

Permutation formula

Permutations are all about arranging things. The key idea is that the order of items matters. The formula to calculate permutations, called \(_{n} P_{r}\), is \(\frac{n!}{(n - r)!}\). This tells us how many ways \(r\) items can be arranged from a set of \(n\) items.
In our example, calculating \(_{5} P_{5}\) involves arranging all 5 items from a set of 5. Using the formula, first calculate \(n!\), which is \(5! = 120\). Second, calculate \((n-r)!\), which is \(0! = 1\). Finally, use the permutation formula: \(_{5} P_{5} = \frac{120}{1} = 120\).
This shows there are 120 different ways to arrange 5 items.

nCr vs nPr

Understanding the difference between combinations (\(_{n} C_{r}\)) and permutations (\(_{n} P_{r}\)) is crucial. - Permutations: In permutations, the order matters. \(_{n} P_{r}\) gives the number of ways to arrange \(r\) items from \(n\) total.- Combinations: In combinations, the order doesn't matter. \(_{n} C_{r}\) is used to find out how many ways \(r\) items can be selected from \(n\), without considering the order.
The formulas are: - Permutation: \(_{n} P_{r} = \frac{n!}{(n-r)!}\)- Combination: \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\)
Remember, the extra \(r!\) in the combination formula adjusts for the fact that different orders of the same items are not considered distinct in combinations.

Counting principles

Counting principles are essential for determining how to count arrangements or selections thoroughly.
- **Fundamental Principle of Counting**: If one event can happen in \(m\) ways, and a second event can happen in \(n\) ways, then both events can happen in \(m \times n\) ways. This principle helps break down complex counting problems into simpler, more manageable parts.
- **Addition Principle**: If an event can happen in one of two mutually exclusive ways, the total number of ways the event can happen is the sum of the number of ways each event can happen. These principles underpin the logic of permutations and combinations, helping to break down complex problems into simpler parts. By applying these principles, students can tackle various counting problems logically and systematically, ensuring they count all possible arrangements or selections.