Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.

a. If the subjects are selected and treated one at a time in sequence, how many different sequential arrangements are possible if 14 people are selected from the 16 that are available?

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Step 1: Recognize that this is a permutation problem because the order in which the 14 subjects are selected and treated matters. In permutations, the arrangement of items is important.

Step 2: Use the formula for permutations, which is given by P(n, r) = n! / (n - r)!, where n is the total number of items (16 volunteers in this case) and r is the number of items to be arranged (14 subjects to be selected).

Step 3: Substitute the values into the formula. Here, n = 16 and r = 14, so the formula becomes P(16, 14) = 16! / (16 - 14)!.

Step 4: Simplify the denominator. Since (16 - 14)! = 2!, the formula becomes P(16, 14) = 16! / 2!.

Step 5: To compute the result, expand the factorials. For example, 16! = 16 × 15 × 14 × ... × 1, and 2! = 2 × 1. Divide 16! by 2! to find the total number of different sequential arrangements possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In this context, it helps determine how many ways we can select and arrange a subset of individuals from a larger group. Specifically, it involves understanding permutations and combinations, which are essential for solving problems related to selecting and ordering subjects in clinical trials.

Permutations

Permutations refer to the different ways in which a set of items can be arranged in order. When selecting 14 subjects from 16, the order in which they are treated matters, making this a permutation problem. The formula for permutations is n! / (n - r)!, where n is the total number of items to choose from, and r is the number of items to arrange.

Factorial

A factorial, denoted by n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is crucial for calculating the total number of arrangements when selecting and ordering subjects in a clinical trial.

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