Problem 57 - Algebra 2 and Trigonometry

Question

A medical researcher has received approval to test a new combination drug therapy. A total of 6 doses of drug \(\mathrm{X}, 3\) doses of drug \(\mathrm{Y},\) and 4 doses of drug \(\mathrm{Z}\) are to be given on successive days. The researcher wants to determine if some orders are more effective than others. How many different orders are possible?

Step-by-Step Solution

Answer

There are 60,060 different possible orders for administering the drugs.

Step 1: Understanding the Problem

The task is to find out how many different ways we can order the doses of three drugs: 6 doses of drug X, 3 doses of drug Y, and 4 doses of drug Z to be administrated on successive days. This is a permutation problem with repetition.

Step 2: Total Doses Calculation

First, calculate the total number of doses to be given which is the sum of all doses of drugs X, Y, and Z. \[6 + 3 + 4 = 13\]

Step 3: Apply Permutation Formula with Repetition

Since we have repetitions in the doses for each drug, use the permutation formula with repetition. The general formula is:\[\frac{n!}{n_1! \times n_2! \times n_3!}\]where \(n\) is the total number of items to arrange (13 in this case), and \(n_1\), \(n_2\), and \(n_3\) are the frequencies of the repeated items. Substitute the frequencies for drugs X, Y, and Z:\[\frac{13!}{6! \times 3! \times 4!}\]

Step 4: Calculate Factorials

Calculate the factorials:- \(13! = 6227020800\)- \(6! = 720\)- \(3! = 6\)- \(4! = 24\)

Step 5: Compute the Number of Orders

Substitute the factorials into the permutation formula:\[\frac{6227020800}{720 \times 6 \times 24} = \frac{6227020800}{103680} = 60060\]

Step 6: Conclusion

There are 60,060 different orders to administer the drugs according to the dose plan given.

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in mathematics, specifically in counting principles and permutations. They are represented by the exclamation mark symbol (!). The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). For example, the factorial of 4, written as \( 4! \), is calculated as \( 4 \times 3 \times 2 \times 1 = 24 \).
Factorials are crucial when calculating permutations, particularly with repetition. In our drug dosage scenario, each type of drug dose can be thought of as repeating items. The overall arrangement involves computing one giant factorial of all items combined, divided by the factorial of each type's count. This division helps to account for the repetition and avoid overcounting. In the problem, 13 doses in total lead to a factorial of 13, and each subgroup (Drug X, Drug Y, and Drug Z) has its own factorial calculated.
Understanding how factorials work makes it easier to handle complex problems involving permutations with repetition, as they are often used to divide overwhelming possibilities into manageable calculations.

Introduction to Combinatorics

Combinatorics is an area of mathematics focused on counting, arranging, and finding patterns. It's an essential tool when dealing with permutations, like in the drug dosage example. In simple terms, combinatorics helps determine how different sets can be combined or arranged in various circumstances. It's often used alongside probability, as it helps calculate the likelihood of certain arrangements or outcomes.
In our example, combinatorics assists in figuring out the possible arrangements of drug dosages. These arrangements are called permutations, especially when order matters. The permutations with repetition formula is a key part of combinatorics and helps when identical items need to be arranged. Here, combinatorics guides us to account for the distinct arrangements while repeating certain items, using the factorial calculations we previously discussed. This approach, of factoring in every identical grouping, ensures accurate results in finding all possible orders.

Drug Dosage Permutations

When it comes to drug dosage permutations, especially in medical research, accurate calculation of possible orders can be crucial. Consider a scenario where different drug doses are administered over several days. If these drugs involve different combinations, knowing the permutation allows researchers to plan and ensure the therapies' effectiveness.
In the given example, we have 6 doses of Drug X, 3 doses of Drug Y, and 4 doses of Drug Z. Permutations with repetition are used because the same type of drug is administered multiple times. By using the formula for permutations with repetition, researchers can calculate 60,060 different possible ways to schedule these doses. This number is a result of both the total number of doses and taking into account the repetition within each dose type. Breaking things down this way helps researchers choose the most effective orders, testing if variations in sequence produce different outcomes, while also ensuring there is balance and scientific rigor in their approach.