Final answer:
The police captain can select any three detectives from seven in 35 different ways, using the combination formula 7C3.
Step-by-step explanation:
The question at hand requires using combinatorial mathematics to determine in how many ways a police captain can choose any three of his seven detectives for a special assignment. To find this, we can use the combination formula which is defined as nCr = n! / [(n-r)! r!], where 'n' is the total number of items, and 'r' is the number of items to choose.
Here, n is 7 (the total detectives) and r is 3 (the detectives needed for the assignment), so we have:
7C3 = 7! / [(7-3)! 3!] = 7! / (4! 3!) = (7*6*5)/(3*2*1) = 35 ways.
Therefore, the police captain can choose any three detectives from his team of seven in 35 different ways.