Problem 3 - College Algebra

Question

In Exercises \(1-8,\) 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: Substitute into the formula

Firstly, we substitute n and r into the permutation formula:\n\(_{8} P_{5} = \frac{8!}{(8-5)!}\)

Step 2: Calculate the factorial

Now we have to calculate the factorial for both 8 and 3 (as 8-5=3):\n8! = 8*7*6*5*4*3*2*1 = 40320,\n3! = 3*2*1 = 6.

Step 3: Finish the calculation

Finally, we divide 8! by (8-5)!:\n\(_{8} P_{5} = \frac{40320}{6} = 6720

Key Concepts

Factorials

Factorials are fundamental in permutations and combinatorics. A factorial, represented with an exclamation mark, such as \(n!\), is a product of all positive integers from 1 to \(n\). For example, \(4!\) means \(4 \times 3 \times 2 \times 1 = 24\). Factorials grow very quickly, making them useful for calculating large numbers of arrangements or sequences. They're easy to understand with practice: just multiply the number by all of the numbers below it down to 1.
Factorials help in finding different orders for arranging items. For instance, if you have 5 books and want to know how many different ways to arrange them on a shelf, you compute \(5!\). This means there are 120 possible arrangements for those books.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of elements. It helps us figure out the number of possible arrangements in set formations, such as permutations and combinations. This field is essential for solving problems related to probabilities, games, and even daily decision-making scenarios.
In combinatorics, it's vital to understand the difference between permutations and combinations. Permutations consider the order of elements, while combinations do not. For instance, arranging three letters \(A, B, C\) in different ways: ABC, ACB, BAC, etc., refers to permutations. However, selecting two out of three, such as AB or BC, are combinations. Combinatorics provides a structured way to calculate how many possible ways there are to arrange or select items.

Permutation Formula

The permutation formula is used to find the number of ways to arrange a subset of items from a larger set, where the order matters. The formula is expressed as \[ _{n} P_{r} = \frac{n!}{(n-r)!} \]This formula helps us calculate cases where the arrangement sequence is crucial, like seating arrangements, race placements, or any scenario where order affects the outcomes.
For example, using \( _{8} P_{5} \), we're calculating the number of ways to arrange 5 people out of 8 in a sequence. This is done by first calculating \(8!\) for the total number of permutations and dividing it by \((8-5)!\) to account for the unused items. This results in \(8!\) divided by \(3!\), giving the number of possible arrangements for this specific subset. Understanding this formula simplifies complex arrangement problems in everyday scenarios and advanced math studies.