Final answer:
To calculate the number of different lineups possible when selecting 4 songs from 7, one can use the combinations formula, C(n, k), resulting in 35 different lineups.
Step-by-step explanation:
The student asks about the number of different lineups possible when selecting 4 songs from a list of 7 songs to compose an event's musical entertainment lineup. This sort of problem falls within the realm of combinatorics, which is a branch of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. The specific mathematical concept to use here is combinations.
When determining the number of combinations, we want to find how many ways we can select items where the order does not matter. The formula for combinations is given by:
C(n, k) = n! / (k! * (n-k)!)
where:
- n is the total number of items to choose from,
- k is the number of items to choose,
- n! denotes n factorial, which is the product of all positive integers up to n,
- k! denotes k factorial, and
- (n-k)! denotes the factorial of the difference between n and k.
In this case, we have 7 songs and we want to choose 4, so our values are n=7 and k=4. Applying the formula:
C(7, 4) = 7! / (4! * (7-4)!) = 7*6*5 / (3*2*1) = 35
Therefore, there are 35 different possible lineups for the musical entertainment.