A teacher has preselected some students to do a sports activity. Since everyone meets the necessary requirements, the teacher will randomly choose only a group of 3 students and finds that he can make 10 possible selections. How many students make up the preselected group?
Let n be the number of students in the preselected group.
We start by calculating the number of possibilities to choose 3 students out of n, which is given by the binomial coefficient C(n, 3) = n!/(3!(n-3)!).
We know that C(n, 3) = 10, so we have the equation n!/(3!(n-3)!) = 10.
Simplifying the equation, we get n!(n-3)!/(3!(n-3)!) = 10, which further simplifies to n!(n-3)! = 10 * 3!.
Since 3! = 6, the equation becomes n!(n-3)! = 60.
We can rewrite 60 as 2^2 * 3 * 5, which means that n! and (n-3)! must include factors of 2, 3, and 5.
The only way for n! and (n-3)! to fulfill this condition is if n = 6 and (n-3)! = 5.
Therefore, the preselected group is made up of 6 students.