Problem 59 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$P(5,2)$$

Step-by-Step Solution

Answer

20

Step 1 - Understand the Notation

The notation \( P(n, r) \) represents the number of permutations of \( n \) objects taken \( r \) at a time.

Step 2 - Apply the Permutation Formula

The formula for permutations is given by \[ P(n, r) = \frac{n!}{(n-r)!} \]. Here \( n = 5 \) and \( r = 2 \). So, apply the values to the formula.

Step 3 - Calculate Factorials

Calculate \( n! \) and \( (n-r)! \). For \( n = 5 \), \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]. For \( n - r = 3 \), \[ 3! = 3 \times 2 \times 1 = 6 \].

Step 4 - Compute the Permutation

Plug the factorial values into the permutation formula: \[ P(5,2) = \frac{5!}{(5-2)!} = \frac{120}{6} = 20 \].

Key Concepts

Permutation Formula

Permutations are a central concept in combinatorics. They refer to the different ways in which a set of objects can be arranged. The permutation formula helps us calculate the number of possible arrangements. This formula is given by \( P(n, r) = \frac{n!}{(n-r)!} \). Here, \( n \) represents the total number of items, and \( r \) is the number of items we are choosing.
Using the formula can seem a bit confusing at first, but it is quite straightforward. It takes into account not just the selection of items but also the order in which they are arranged. This is different from combinations where order doesn't matter. When you see a problem involving permutations, remember: it's all about the order!

Factorials

Factorials are another vital concept when dealing with permutations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For instance:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials grow very quickly with larger numbers. They are used in calculations of permutations because they account for all possible orderings of a set of items. In the permutation formula, both \( n! \) and \( (n-r)! \) involve factorials, making this concept essential to understand.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, both as a means and an end in obtaining results. It includes the study of permutations, combinations, and other techniques of counting.
In our example with \( P(5,2) \), we're using combinatorics to find the total arrangements of 5 items taken 2 at a time. Combinatorics allows us to explore all possibilities systematically rather than listing them out manually, which can be inefficient and error-prone. Understanding basic principles of combinatorics, like permutations and combinations, sets a strong foundation for tackling more complex problems in probability and statistics.

Step by Step Solutions

Breaking down complex problems into simpler steps is a highly effective approach in mathematics. Let's revisit the solution for \( P(5,2) \) step by step:

• Step 1 - Understand the Notation: Recognize what \( P(n, r) \) means - permutations of \( n \) items taken \( r \) at a time.


• Step 2 - Apply the Permutation Formula: Use the formula \( P(n, r) = \frac{n!}{(n-r)!} \). Identify \( n = 5 \) and \( r = 2 \).


• Step 3 - Calculate Factorials: Find \( 5! \) and \( 3! \) because \( (n-r) = 3 \). So, \( 5! = 120 \) and \( 3! = 6 \).


• Step 4 - Compute the Permutation: Substitute the factorials into the formula: \( P(5, 2) = \frac{120}{6} = 20 \).

Following these structured steps helps in understanding and solving permutation problems easily. This process can be applied to any permutation problem, making it a useful strategy in combinatorics.