Tell whether each situation involves combinations or permutations. Then give the number of possible outcomes. A manager is scheduling interviews for job applicants. She has 8 time slots and 6 applicants to interview. How many different interview schedules are possible?
One of the main differences between a combination and a permutation is that
the order matters in a permutation.
e.g. When picking a committee of 6 people from 8 men and women, it does not matter in which order you pick them, that is, C(8,6) = 8!/(6!2!) = 8*7/2 = 28 , but...
if you were to pick a president, vice president and secretary from 5 applicants, their position makes it a permutation, that is, P(5,3) = 5!/2! = 5*4*3 = 60
In your case, it does matter how the time slots are filled with different people, you don't want applicants to show up at the wrong time, or be late, so
think of it as the 6 applicants being assigned times.
the first one can be given 8 different times,
leaving 7 different times for the 2nd applicant,
leaving 6 different times for the 3rd applicant,
....
leaving 1 time slot for the 6th applicant, so we have 8*7*6*5*4*3 = P(8,6) = 20160
This situation involves permutations, as the order in which the applicants are scheduled for interviews matters.
To calculate the number of possible outcomes, we need to calculate the number of ways we can arrange 6 applicants in the 8 available time slots.
We can use the formula for permutations to calculate this:
P(n, r) = n! / (n - r)!
Where n is the total number of items to choose from, and r is the number of items to be selected.
In this case, we have 6 applicants (n) and 8 time slots (r).
P(6, 8) = 6! / (6 - 8)!
= 6! / (-2)!
= 6! / 2!
= (6 × 5 × 4 × 3 × 2 × 1) / (2 × 1)
= 720 / 2
= 360
Therefore, there are 360 different interview schedules possible.
To determine whether this situation involves combinations or permutations, we need to consider the order in which the applicants are scheduled.
Since the order of the applicants does not matter (i.e., two different schedules with the same applicants interviewed in different orders are considered the same), this situation involves combinations.
To calculate the number of different interview schedules, we need to use the combination formula, which is given by:
C(n, r) = n! / (r!(n-r)!)
In this case, we have 8 time slots (n) and 6 applicants (r). Plugging in the values, we get:
C(8, 6) = 8! / (6!(8-6)!)
Simplifying further:
C(8, 6) = (8 * 7 * 6!) / (6! * 2!)
Canceling out the common terms (6!):
C(8, 6) = 8 * 7 / 2
C(8, 6) = 28
Therefore, there are 28 different interview schedules possible for the manager to choose from.