Mark wants to visit the 10 colleges he is considering attending. He can only spend the night at 3 of them. What is the probability that he spends a night at Rutgers University, a night at the University of Miami, and a night at Clemson University?
number of triples from 10
= C(10,3) = 120
the triple of Rutgers, Miami, and Clemson is specifically one of those
prob(your event) = 1/120
thanks
To calculate the probability that Mark spends a night at Rutgers University, the University of Miami, and Clemson University, we need to know the total number of possible combinations.
Mark has 10 colleges to choose from, and he can only spend the night at 3 of them. Therefore, the total number of combinations is given by the binomial coefficient formula, also known as "n choose k," where n is the total number of options, and k is the number of options chosen:
C(10, 3) = 10! / (3!(10-3)!) = 120 / (6 * 5040) = 120 / 720 = 1/6
So, the probability that Mark spends a night at Rutgers University, the University of Miami, and Clemson University is 1/6.
To find the probability that Mark spends a night at Rutgers University, a night at the University of Miami, and a night at Clemson University, we need to know the total number of possible outcomes and the number of favorable outcomes.
Since Mark can only spend the night at 3 of the 10 colleges he is considering, the total number of possible outcomes is the number of ways to choose 3 colleges out of 10, which can be calculated using the combination formula.
The combination formula is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of items and r is the number of items chosen.
For our case, n = 10 (total number of colleges) and r = 3 (number of nights Mark can spend).
Using the formula, we can calculate the total number of possible outcomes as:
C(10, 3) = 10! / (3! * (10 - 3)!)
= 10! / (3! * 7!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
Now, in order to spend a night at Rutgers University, a night at the University of Miami, and a night at Clemson University, Mark needs to choose these three specific colleges out of the 10 available colleges.
The number of favorable outcomes is 1, as there is only one way to choose these three specific colleges out of 10.
Therefore, the probability that Mark spends a night at Rutgers University, a night at the University of Miami, and a night at Clemson University is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 1 / 120
= 0.0083 (rounded to four decimal places)
= 0.83% (rounded to the nearest whole number)
So, the probability that Mark spends a night at Rutgers University, a night at the University of Miami, and a night at Clemson University is approximately 0.83% or 1 in 120.