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 a b , where a and b are positive, coprime numbers. What is the value of a+b ?
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To find the probability, we first need to calculate the total number of ways Bibi can choose 21 friends out of the 37 she has. This can be calculated using combinations.
The number of ways to choose 21 friends out of 37 is given by the combination formula:
C(37, 21) = 37! / (21!(37-21)!)
Next, we need to calculate the total number of ways the message can be sent to three random contacts out of the 37. This can be calculated using combinations as well.
C(37, 3) = 37! / (3!(37-3)!)
Now, to find the probability that exactly one person who got the invitation was not supposed to be invited to the party, we need to divide the number of favorable outcomes (where one person not invited is chosen) by the total number of possible outcomes.
The number of favorable outcomes is choosing 1 person out of the 21 supposed invitees and 2 people out of the remaining 16 non-invited people. This can be calculated using combinations as well.
C(21, 1) * C(16, 2) = (21! / (1!(21-1)!)) * (16! / (2!(16-2)!))
Finally, we can calculate the probability:
P = (C(21, 1) * C(16, 2)) / C(37, 3)
So, the probability can be expressed as a fraction. To find the value of a+b, we need to find the numerator and denominator of the fraction separately. Then, we can add the two numbers.
Numerator = C(21, 1) * C(16, 2)
Denominator = C(37, 3)
Now, let's calculate the values of the numerator and denominator.
Numerator = (21! / (1!(21-1)!)) * (16! / (2!(16-2)!))
Denominator = 37! / (3!(37-3)!)
After calculating the values, we can find a+b by finding the sum of the numerator and denominator.
a+b = Numerator + Denominator