Problem 8 - College Algebra

Question

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{6} P_{0} $$

Step-by-Step Solution

Answer

The solution to \(_{6} P_{0}\) is 720.

Step 1: Interpret the problem in terms of permutations

This problem asks us to evaluate \(_{6} P_{0}\). This is the number of ways to arrange 6 items in 0 spaces.

Step 2: Apply the formula for \(_{n} P_{r}\)

The general formula for a permutation is \(_{n} P_{r} = n! / (n-r)!. For our problem, this becomes \(_{6} P_{0} = 6! / (6-0)!\)

Step 3: Calculate the factorials

6! is 720 (6*5*4*3*2*1), and \(0!=1\) by definition. So the formula becomes \(720 / 1\).

Step 4: Evaluate the expression

When you divide 720 by 1, the solution to \(_{6} P_{0}\) is 720.

Key Concepts

Factorials

Factorials are a foundation of permutations. A factorial, represented by an exclamation mark "!", is the product of an integer and all the integers below it. For example:

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

Factorials are used to calculate permutations and combinations. Important to note is that by definition, \(0!\) is always 1. This definition ensures that mathematical formulas work smoothly even when accommodating zero elements.
Understanding how to calculate factorials is crucial because they map out all possible arrangements for a set number of items.

Permutation Formula

The permutation formula, written as \(_n P_r\), allows us to determine the number of ways to arrange "n" distinct items into "r" spaces.
The formula is:\[ _n P_r = \frac{n!}{(n-r)!} \]This sees us utilizing factorial mathematics to achieve an order-specific selection.

• The numerator \(n!\) calculates the total permutations for "n" items.
• The denominator \((n-r)!\) reduces this number down by accounting for the positions not being used.

For example, in solving \(_6 P_0\), the formula becomes \(\frac{6!}{(6-0)!}\). This effectively simplifies to 1, the number of ways to arrange zero items.

Arrangements

In permutations, we focus on arrangements, where the order of items significantly matters. Arrangements reflect how various elements can be positioned or ordered. With permutations, we can explore several examples:

• If you have 3 books and want to arrange all of them, the sequence would matter. Thus, the arrangement is calculated using permutations.
• When calculating \(_6 P_0\), asking for arrangements of zero spaces with six items might initially seem tricky. However, it enlightens us that there is precisely one way to arrange nothing, i.e., doing nothing (which is itself an arrangement).

These concepts allow us to embrace the depth of organization and order, which influences countless real-world applications, from scheduling to optimization.