Bibi wants to send invitations to some of her friends to come to her birthday party. She has contact information for 37 friends stored on her phone, but she only wants to invite 21 of those people to her party. She tries to send out a message to the 21 people, but her phone malfunctions and sends the message to three random contacts. The probability that exactly one person who got the invitation was not supposed to be invited to the party can be expressed as ab, where a and b are positive, coprime numbers. What is the value of a+b?
To find the probability that exactly one person who got the invitation was not supposed to be invited to the party, we can use the concept of combinations.
The total number of ways to select 3 random contacts from the 37 friends is given by the combination formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of friends and r is the number of contacts we want to send the message to.
In this case, n = 37 and r = 3, so the total number of ways to select 3 random contacts from the 37 friends is:
C(37, 3) = 37! / (3!(37-3)!) = 37! / (3!34!)
To find the probability of sending the message to exactly one person who was not supposed to be invited, we need to calculate the following:
(Number of ways to select 1 of the 21 invited friends) * (Number of ways to select 2 of the remaining 16 friends who were not invited) / (Total number of ways to select 3 random contacts)
Number of ways to select 1 of the 21 invited friends = C(21, 1) = 21
Number of ways to select 2 of the remaining 16 friends who were not invited = C(16, 2) = 16! / (2!(16-2)!) = 16! / (2!14!) = 8 * 15 = 120
Therefore, the probability is:
(21 * 120) / (37! / (3!34!)) = (21 * 120 * (3!34!)) / 37!
Now, let's calculate the numerical value of this probability to express it as ab:
Probability = (21 * 120 * (3!34!)) / 37!
The value of a is 21 * 120 * (3!34!) = 14,784, and b is 37!.
Therefore, a + b = 14,784 + 37!
To find the probability that exactly one person who got the invitation was not supposed to be invited to the party, we need to calculate two separate probabilities:
1. The probability of sending the message to exactly one person who was not supposed to be invited.
2. The probability of sending the message to exactly one person who was supposed to be invited.
Let's start with the first probability. There are three random contacts who received the message out of the 37 friends stored on Bibi's phone. We need to choose one person who was not supposed to be invited from the 16 remaining friends (37 - 21). The other two contacts must be among the remaining 21 friends.
The probability of choosing one person who was not supposed to be invited is (16/37), and the probability of choosing two people who were supposed to be invited is (21/36) * (20/35). Since order doesn't matter, we multiply this by the number of possible combinations of choosing one person who was not supposed to be invited out of the three random contacts, which is (3 choose 1) = 3.
The first probability can be calculated as: (16/37) * (21/36) * (20/35) * 3.
For the second probability, we need to choose one person who was supposed to be invited from the 21 friends invited to the party. The other two contacts must be among the remaining 16 friends (37 - 21).
The probability of choosing one person who was supposed to be invited is (21/37), and the probability of choosing two people who were not supposed to be invited is (16/36) * (15/35). Again, we multiply this by the number of possible combinations of choosing one person who was supposed to be invited out of the three random contacts, which is again (3 choose 1) = 3.
The second probability can be calculated as: (21/37) * (16/36) * (15/35) * 3.
Now, to find the probability that exactly one person who got the invitation was not supposed to be invited, we need to sum up these two probabilities:
Probability = (16/37) * (21/36) * (20/35) * 3 + (21/37) * (16/36) * (15/35) * 3.
Calculating this probability will give us a fraction in the form ab, where a and b are positive, coprime numbers. The sum of a+b will be the answer to the question.