Question

medications for depression a researcher wishes her patients to try a new medicine for depression. how many different ways can she select 5 patients from 41 patients? there are \square different ways to select the patients.

Explanation:

Step1: Identify the problem type

This is a combination problem where we need to find the number of ways to choose 5 patients from 41, which is given by the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 41 \) and \( k = 5 \).

Step2: Apply the combination formula

First, calculate the factorials. The formula becomes \( C(41, 5)=\frac{41!}{5!(41 - 5)!}=\frac{41!}{5!×36!} \). Since \( n!=n\times(n - 1)\times\cdots\times(n - k+1)\times(n - k)! \), we can simplify \( \frac{41!}{36!}=41\times40\times39\times38\times37 \). Then, \( 5!=5\times4\times3\times2\times1 = 120 \).

Step3: Calculate the numerator and denominator

Calculate \( 41\times40\times39\times38\times37 \):
\( 41\times40 = 1640 \)
\( 1640\times39 = 63960 \)
\( 63960\times38 = 2430480 \)
\( 2430480\times37 = 89927760 \)

Then divide by \( 5! = 120 \): \( \frac{89927760}{120}=749398 \)

Answer:

749398