Final answer:
There are 3060 ways to choose four freshmen from a group of eighteen to be members of the leadership board, calculated using the combinations formula in combinatorics.
Step-by-step explanation:
To determine how many ways four freshmen can be chosen from eighteen to be members of the leadership board, we can use the concept of combinations in probability and combinatorics. This is a type of counting problem where the order of selection does not matter.
In combinatorial mathematics, the number of ways to choose 'r' elements from a set of 'n' elements without regard to the order of selection is given by the formula for combinations, which is:
C(n, r) = n! / (r!(n - r)!)
Where 'n' is the total number of items to choose from, 'r' is the number of items to choose, and '!' represents factorial, which is the product of all positive integers less than or equal to a given positive integer.
Using the formula:
C(18, 4) = 18! / (4!(18 - 4)!) = 18! / (4!14!) = (18 × 17 × 16 × 15) / (4 × 3 × 2 × 1) = 3060
Therefore, there are 3060 ways to choose four freshmen from eighteen to be members of the leadership board.