Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{6} P_{5}$$

Step-by-Step Solution

Answer

The permutation of 5 items taken from a pool of 6 distinct items, _{6}P_{5}, is 720.

Step 1: Understanding the permutation formula

The permutation formula is represented as _{n}P_{r} = \( \frac{n!}{(n-r)!} \). Here, 'n' is the total number of items, 'r' is the items we are selecting, and '!' is the factorial operator. Factorial of a number, say 'n', is the product of all positive integers less than or equal to 'n'.

Step 2: Substitute the values into the permutation formula

Substitute 'n' with 6, and 'r' with 5 into the permutation formula. It results in \( \frac{6!}{(6-5)!} \)

Step 3: Solve for Factorials

First calculate factorials for the numbers in the numerator and denominator. So, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720, and 1! = 1.

Step 4: Divide to get final answer

To get the final answer, divide 720 by 1 which equals 720.

Key Concepts

Factorial

The concept of a factorial is quite straightforward, yet incredibly powerful in mathematics, especially in the field of combinatorics. A factorial is denoted by an exclamation mark ! and is applicable to positive integers. For instance, when we see the expression 6!, it implies that we must multiply a series of descending natural numbers: 6 x 5 x 4 x 3 x 2 x 1.
This operation results in 720. Hence, 6! equals 720.
Factorials are essential when calculating permutations and combinations because they help us count and arrange objects in specific orders. Key points to remember about factorials include:

• Zero factorial, or 0!, is defined as 1.
• Factorials grow very large, very quickly as the number increases.
• Useful in many areas of math like probability, calculus, and algebra.

Permutation formula

The permutation formula is a fundamental concept in combinatorics. It helps us determine how many ways we can arrange a certain number of items where the order matters. This makes permutations distinct from combinations, where order is not a factor.
The formula for permutations is expressed as:\[_{n}P_{r} = \frac{n!}{(n-r)!}\]Here, \(n\) represents the total number of available items, while \(r\) symbolizes the number of items you intend to arrange. The "!" symbol signifies a factorial operation.
To use this formula, you:

• Calculate the factorial of the total number of items \(n!\).
• Then, calculate the factorial of the difference \((n-r)!\).
• Finally, divide the factorial of \(n\) by the factorial of \(n-r\) to get your answer.

In our example, _{6}P_{5}, it means arranging 5 items out of 6, which results in 720 distinct ways.

Combinatorics

Combinatorics is an area of mathematics that studies counting, arrangement, and combination of objects. It provides us with tools to analyze different possible configurations in a systematic way.
Combinatorics is vastly utilized in computer science, probability, geometry, and more. It has various subfields, including:

• Permutations: An arrangement of objects where order matters. For instance, choosing 3 students from a class of 5 and assigning them roles of president, vice-president, and secretary.
• Combinations: A selection of objects where order does not matter, such as selecting 3 fruits from a basket of 5 different ones.
• Graph theory: Studies the relationships between pairs of objects.

Understanding the difference between permutations and combinations is crucial in combinatorics, as it determines whether or not the order of arrangement impacts the total number of outcomes. In permutation problems, such as our example _{6}P_{5}, the sequence matters, therefore, each different arrangement counts as a separate permutation.