There are 14 qualified applicants for 3 trainee positions in a fast-food management program. How many different groups of trainees can be selected?
To calculate the number of different groups of trainees that can be selected, we can use the combination formula.
The formula for calculating combinations is:
C(n, r) = n! / (r! * (n - r)!)
Where:
- n is the total number of applicants (14 in this case)
- r is the number of trainees to be selected (3 in this case)
Using the numbers from your question, the calculation would be as follows:
C(14, 3) = 14! / (3! * (14 - 3)!)
= 14! / (3! * 11!)
First, let's calculate the factorial of 14:
14! = 14 * 13 * 12 * 11!
Next, let's calculate the factorial of 3:
3! = 3 * 2 * 1
Finally, let's calculate the factorial of 11:
11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Now we can substitute these values in the formula:
C(14, 3) = (14 * 13 * 12 * 11!) / (3 * 2 * 1 * 11!)
The factorials in the numerator and denominator cancel out, leaving us:
C(14, 3) = 14 * 13 * 12 / (3 * 2 * 1)
C(14, 3) = 364
Therefore, there are 364 different groups of trainees that can be selected from the 14 qualified applicants.
To find the number of different groups of trainees that can be selected, we can use the concept of combinations.
In this case, we need to select 3 trainees from a pool of 14 qualified applicants. The order in which the trainees are selected does not matter, so we are looking for combinations, not permutations.
The formula for combinations is nCr, where n represents the total number of items and r represents the number of items to be selected.
In this case, n = 14 (total number of applicants) and r = 3 (number of trainees to be selected).
The formula for combinations is:
nCr = n! / (r!(n-r)!)
Using this formula, we can find the number of different groups of trainees that can be selected:
nCr = 14! / (3!(14-3)!)
= 14! / (3!11!)
Calculating the factorial values:
14! = 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
3! = 3 x 2 x 1
11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Now, we can substitute the factorial values into the formula:
nCr = (14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((3 x 2 x 1)((11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1))
Calculating the expression:
nCr = (14 x 13 x 12) / (3 x 2 x 1)
= (2184) / (6)
= 364
Therefore, there are 364 different groups of trainees that can be selected from the 14 qualified applicants for the 3 trainee positions in the fast-food management program.