Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{8} P_{3}$$

Step-by-Step Solution

Answer

Therefore, \( _{8} P_{3} = 336 \).

Step 1: Identify n and r

We are to solve for \( _{8} P_{3} \), meaning \( n = 8 \) and \( r = 3 \).

Step 2: Substitute n and r into the formula

Substitute \( n \) and \( r \) into the permutation formula. It becomes \( \frac{8!}{(8 - 3)!} \). Now simplify the equation.

Step 3: Simplify the equation

Simplify the equation to \( \frac{8!}{5!} \). Calculating the factorials gives \( \frac{40320}{120} \).

Step 4: Perform the division

Dividing 40320 by 120 gives 336.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics that demonstrate how to calculate the product of all positive integers up to a given number. This is denoted by an exclamation point. For example, the factorial of 5, written as 5!, means 5 multiplied by every positive whole number below it. Hence, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Understanding factorials is crucial for solving permutations and combinations since these mathematical tools rely heavily on the repetitive product of sequential numbers.
Note also that 0! is defined as 1. This might seem strange, but it's used to maintain the mathematical consistency of arrangements and calculations.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing sets of elements. It's particularly useful for determining the number of ways elements can be selected or organized.
For a practical example, consider trying to find out how many ways you can arrange a set of books or select items for a committee.
Combinatorics helps solve such problems by utilizing concepts like permutations, combinations, and the pigeonhole principle.

• Permutations: Arranging items where order matters.
• Combinations: Selecting items where order does not matter.
• Pigeonhole Principle: Proving the inevitability of certain arrangements.

Mastery of these concepts provides a strong foundation to tackle various mathematical permutations and combinations problems.

Permutation Formula

The permutation formula is a mathematical expression used to determine the number of arrangements of a subset of items. This formula is applicable when the order of arrangement is important. The formula is denoted by \( _{n} P_{r} = \frac{n!}{(n - r)!} \), where:

• \( n \) is the total number of items.
• \( r \) is the number of items to arrange.

To solve a permutation problem like \( _{8} P_{3} \), you substitute the values of \( n \) and \( r \) into the formula:
\[ _{8} P_{3} = \frac{8!}{(8 - 3)!} = \frac{8!}{5!} \]
After calculating the factorials, this reduces to \( \frac{40320}{120} = 336 \).
This result shows there are 336 ways to arrange a group of 3 out of 8 items, where the order matters.