Problem 3 - Algebra and Trigonometry

Question

Use the formula for \(_{n} P_{r}\) to evaluate each expression. \(_{8} P_{5}\)

Step-by-Step Solution

Answer

The value of \(_{8} P_{5}\) is 6720.

Step 1: Identify the values of n and r

In this task, the given expression is \(_{8} P_{5}\). This implies that \(n = 8\) and \(r = 5\).

Step 2: Substitute the values of n and r in the permutation formula

According to the permutation formula \(_{n} P_{r} = \frac{n!}{(n-r)!}\), substitute \(n = 8\) and \(r = 5\) into the formula. This results in: \(_{8} P_{5} = \frac{8!}{(8-5)!}\).

Step 3: Evaluate the Factorials

To complete the calculation, we must evaluate the factorials in the formula. We can calculate the factorials as: \(8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320\) and \((8 - 5)! = 3! = 3 × 2 × 1 = 6\). \(_8 P_5 = \frac{40320}{6}\).

Step 4: Complete the Calculation

Completing the division operation, we find: \(_{8} P_{5} = \frac{40320}{6} = 6720\).

Key Concepts

Factorial Notation

Factorial notation is essential for understanding permutations, and it's symbolized with an exclamation point (!). Essentially, the factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n.

For instance, 5! (read as 'five factorial') is calculated by multiplying 5 by every positive integer that comes before it down to 1:
\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].

It's important to remember a unique case: 0! is always equal to 1. This convention makes many mathematical formulas work correctly, including those involving permutations and combinations. When dealing with factorial notation in exercises and solving for permutations, students should evaluate the factorials step by step, ensuring they understand the multiplication sequence involved.

nP-r Notation

The nPr notation expresses the number of permutations of r elements that can be selected from a set of n distinct elements. This concept is fundamental in probability and combinatorics.

The permutation formula is expressed as _n P_r, which reads 'n permute r', and it's calculated using the factorials of n and n−r: \[_n P_r = \frac{n!}{(n-r)!}\].

In this formula, the numerator n! accounts for the total number of ways to arrange n elements, and the denominator (n−r)! adjusts for arranging the remaining n−r elements that are not part of the permutation. This calculation gives the precise number of unique arrangements of r elements selected from a group of n. Understanding this concept allows students to tackle a variety of problems in statistics and combinatorial exercises.

Evaluating Permutations

Evaluating permutations involves applying the nPr notation and factorial concept to determine the number of different ways to arrange a subset of elements.

For example, let's look at the process of evaluating _8 P_5 step by step. First, identify n which is the total number of elements to choose from, and r the number of elements to arrange. In this case, n=8 and r=5. Next, use the permutation formula: \[_8 P_5 = \frac{8!}{(8-5)!}\].

Then, calculate the factorial of n (8!) and (n−r) ((8−5)!) separately before dividing them as per the formula. After performing the calculation (in this case, \[_8 P_5 = \frac{40320}{6} = 6720\]), the result gives the number of ways to arrange 5 elements out of 8. Breaking down the steps to evaluate permutations simplifies understanding how the elements are being rearranged and demystifies the process. By practicing these evaluations, students can quickly become adept at solving permutation-related problems across different contexts.