UpStudy Free Solution:

To determine the number of ways to assign four qualified applicants to ten hospitals, we need to consider that each applicant can be assigned to any of the ten hospitals independently. This is a problem of permutations with repetition.
For each of the four applicants, there are ten choices (hospitals) for assignment. Therefore, the total number of ways to assign the applicants is given by:
\[10 \times 10 \times 10 \times 10 = 10^ 4\]
Calculating \(10^ 4\):
\[10^ 4 = 10,000\]
So, there are 10,000 different ways to assign the four qualified applicants to the ten hospitals.
 

Supplemental Knowledge

In combinatorics, when we need to assign a certain number of items (in this case, nurses) to a larger number of positions (hospitals), we often use permutations or combinations depending on whether the order matters. Here, since each nurse can be assigned to any hospital and the order of assignment matters, we use permutations.
Given:
10 hospitals
4 qualified applicants (nurses)
We need to determine how many ways the 4 nurses can be assigned to the 10 hospitals.
The formula for permutations where \(n\) is the total number of items and \(r\) is the number of items to choose is given by:
\[P( n, r) = \frac { n! } { ( n- r) ! } \]
In this case:
\[n = 10\]
\[r = 4\]
 

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