Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. In how many ways can 7 of 10 chemicals be added to a beaker for an experiment? (Assume that the order in which the chemicals are add is important.)

Step-by-Step Solution

Answer

There are 604,800 ways to add 7 of 10 chemicals to the beaker.

Step 1: Identify the Problem Type

This problem involves selecting a subset of chemicals from a larger set, where the order of selection matters. Therefore, it is a permutation problem.

Step 2: Understand Permutation Formula

The formula for permutation where order matters is given by \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.

Step 3: Apply the Permutation Formula

For this problem, we need to choose 7 chemicals from a set of 10, so \( n = 10 \) and \( r = 7 \). Plug these values into the permutation formula: \[ P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!} \]

Step 4: Calculate the Permutations

Compute the factorial values: \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \) \( 3! = 3 \times 2 \times 1 = 6 \)Now substitute back into the equation:\[ P(10, 7) = \frac{3,628,800}{6} = 604,800 \]

Step 5: Finalize the Solution

Thus, the number of ways to add 7 chemicals out of 10 to a beaker, considering the order of addition, is \( 604,800 \).

Key Concepts

Fundamental Principle of Counting

At its core, the Fundamental Principle of Counting is a straightforward concept used in combinatorics. It helps us determine the number of possible ways events can occur. Simply put, if one event can happen in "m" ways and a second event can happen independently in "n" ways, the two events can occur in a total of \( m \times n \) ways.

• For multiple events, you simply multiply the number of possible outcomes of each event.
• For instance, if you're picking a shirt and pants for an outfit, and you have 3 shirts and 4 pants to choose from, you have \( 3 \times 4 = 12 \) possible outfits.

This principle is foundational in understanding more complex counting methods, such as permutations and combinations. By applying this principle, we can break down more complicated problems into manageable parts. When we talk about permutations, we're essentially multiplying choices across sequences, hence building on this fundamental counting rule.

Permutation Formula

Permutations are a way to count arrangements or sequences where order matters. The permutation formula helps us calculate how many different ways a set of items can be arranged.
The formula for permutations is \( P(n, r) = \frac{n!}{(n-r)!} \), where:

• \( n \) is the total number of items.
• \( r \) is the number of items to arrange.
• \( ! \) denotes a factorial, a product of all positive integers up to that number.

The key is that order is important in permutations. If you rearrange the elements, you create a different permutation. For example, the arrangements "ABC" and "ACB" are considered different because the order of elements differs. By applying the permutation formula, you can determine the total number of sequences possible, which is crucial for solving problems where arrangement affects the outcome.

Factorials

Factorials are a mathematical operation often used in permutations and combinations. The notation for a factorial is an exclamation mark "!" following a number. This represents the product of all positive integers up to and including that number. For example:

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

Factorials grow very quickly. They are used mainly to determine permutations and combinations due to their ability to account for sequences and arrangements.
In permutation problems, factorials help us understand the total number of ways to arrange a set of items without repetition. For example, for \( n \) items, arranging all of them is \( n! \). This provides a foundation to solve problems that require counting specific arrangements where order counts.

Order in Combinatorics

The concept of order in combinatorics addresses whether the sequence of items does or does not matter. In permutations, order significantly affects the result.

• Every different arrangement or sequence is considered unique if order matters.
• For example, the sequences "123" and "321" are different when order is considered.

In combinatorics, recognizing whether or not order is significant can affect how you solve a problem. If the order isn't important, you'd use combinations instead of permutations.
However, when order is important, permutations provide the right framework for calculating arrangements. It's crucial to identify this aspect in problems because it influences whether to use permutation formulas like \( P(n, r) = \frac{n!}{(n-r)!} \) or methods for combinations when calculating possible outcomes. Understanding this distinction is key for accurately arranging items in sequences or sets.