here are 12 qualified applicants for 8 trainee positions in a fast-food management program. How many different groups of trainees can be selected?
495
11880
To calculate the number of different groups of trainees that can be selected, we can use the combination formula.
The formula for combination is:
C(n, r) = n! / (r!(n-r)!)
Where:
n = total number of applicants
r = number of positions available (trainee positions)
In this case:
n = 12 (total number of applicants)
r = 8 (number of trainee positions)
Using the combination formula:
C(12, 8) = 12! / (8!(12-8)!)
C(12, 8) = 12! / (8! * 4!)
Simplifying:
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 479,001,600
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
4! = 4 * 3 * 2 * 1 = 24
C(12, 8) = 479,001,600 / (40,320 * 24)
C(12, 8) = 479,001,600 / (967,680)
C(12, 8) ≈ 495
Therefore, there are approximately 495 different groups of trainees that can be selected.
To find the number of different groups of trainees that can be selected, we need to use the concept of combinations.
In this case, we have 12 qualified applicants for 8 trainee positions, which means we need to select 8 trainees from the group of 12 applicants.
The formula for combinations is given by C(n, r) = n! / (r!(n-r)!), where n represents the total number of items and r represents the number of items we are selecting.
In our case, n = 12 (the total number of applicants) and r = 8 (the number of trainees we need to select).
Plugging these values into the formula, we get:
C(12, 8) = 12! / (8!(12-8)!) = 12! / (8! * 4!)
Now, let's calculate the value of 12! (12 factorial):
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 479,001,600
Similarly, let's calculate the value of 8! (8 factorial) and 4! (4 factorial):
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
4! = 4 * 3 * 2 * 1 = 24
Now, substitute these values back into the combinations formula:
C(12, 8) = 479,001,600 / (40,320 * 24)
= 479,001,600 / 967,680
= 495
Therefore, there are 495 different groups of trainees that can be selected from the 12 qualified applicants for 8 trainee positions.