Problem 45 - Algebra for College Students

Question

Evaluate expression. \(\frac{P(10,4)}{4 !}\)

Step-by-Step Solution

Answer

210

Step 1 - Understand the Notation

Understand that \(P(10,4)\) stands for permutations of 10 items taken 4 at a time. This is calculated using the formula \(P(n, k) = \frac{n!}{(n-k)!}\).

Step 2 - Plug in the Values for Permutation

Using the formula \(P(10,4)\), we get \(P(10,4) = \frac{10!}{(10-4)!} = \frac{10!}{6!}\).

Step 3 - Simplify \(10! / 6!\)

Since \(10! = 10 \times 9 \times 8 \times 7 \times 6!\), simplify to \(\frac{10 \times 9 \times 8 \times 7 \times 6!}{6!}\). The \(6!\) cancels out, leaving us with \(10 \times 9 \times 8 \times 7\).

Step 4 - Calculate the Value of the Expression

Now multiply the remaining values: \(10 \times 9 = 90\), \(90 \times 8 = 720\), and \(720 \times 7 = 5040\). Thus, \(P(10, 4) = 5040\).

Step 5 - Consider the Factorial in the Denominator

Recall that the given expression is \(\frac{P(10,4)}{4!}\). We already computed \(P(10,4) = 5040\). Now calculate \(4!\), which is \(4 \times 3 \times 2 \times 1 = 24\).

Step 6 - Complete the Division

Finally, divide the result of the permutation by \(4!\). Compute \(\frac{5040}{24} = 210\).

Key Concepts

Permutations

Permutations show how we can arrange a certain number of items. The order matters in permutations. For example, if you have five books and you want to know the number of ways you can arrange three of them on a shelf, you would use permutations.

Permutations are represented by the symbol \(P(n, k)\). This means the permutation of n items taken k at a time. Mathematically, it's written and calculated as: \[P(n, k) = \frac{n!}{(n-k)!}\].

Here, n! (read as 'n factorial') represents the product of all positive integers from 1 up to n. For example, if we need to calculate the permutations \(P(10, 4)\), we would use: \[P(10, 4) = \frac{10!}{(10-4)!} = \frac{10!}{6!}\]

Therefore, permutations help us when the order of selection is important in problems.

Factorial

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. It's denoted with an exclamation mark (n!).

For example, 4! means: \[4! = 4 \times 3 \times 2 \times 1 = 24\].

Factorials grow very quickly with larger numbers. For instance, 10! looks like this: \[10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\].

Factorials are foundational to many areas of combinatorial mathematics including permutations and combinations.

Combinatorial Mathematics

Combinatorial mathematics involves counting, arranging, and grouping items. Two major concepts are combinations and permutations.

• Permutations: These deal with arrangements where the order matters. For instance, arranging 4 out of 10 books has a specific order, like the math problem we discussed earlier:
  \( P(10,4) \) represents the permutations of 10 items taken 4 at a time, and is calculated by using: \[P(10,4) = \frac{10!}{6!}\].


• Combinations: These deal with groups where the order doesn't matter. For example, picking 3 out of 10 books without worrying about the order they are picked.

These concepts are useful in solving problems like finding probabilities and arrangements.

Evaluating Expressions

Evaluating expressions simply means computing their value step by step.

Let's break down our example: \[\frac{P(10,4)}{4!}\].

• First, we evaluated the permutation: \[P(10,4) = \frac{10!}{6!} = 5040.\]


• Next, we calculated the factorial in the denominator: \[4! = 24.\]


• Finally, we divided the permutation by the factorial: \[\frac{5040}{24} = 210.\]

Step-by-step evaluation helps in simplifying even complex expressions into smaller, manageable parts.