While spending 10 weeks at a university in Germany, Ayesha frequently had dinner with 5 of her classmates: Jack, Erik, Annalee, Peter, and Nola. She ate dinner with each of these classmates exactly 12 times. She ate dinner with every pair of them eight times. She ate dinner with every set of 3 classmates ten times. She ate dinner with every foursome five times. She ate with all five classmates four times. On how many days did Ayesha have dinner with none of these classmates?
10 Weeks = 70 days
12 + 8 + 10 + 5 + 4 = ?
Subtract the above answer from 70.
To find out how many days Ayesha had dinner with none of her classmates, we need to subtract the total number of days she had dinner with at least one classmate from the total number of days during her stay.
First, let's calculate the total number of days Ayesha spent at the university. Since we are given that she spent 10 weeks there, and there are 7 days in a week, we can calculate the total number of days as follows:
10 weeks * 7 days/week = 70 days
Now, let's calculate the total number of days she had dinner with at least one classmate:
She ate dinner with each classmate 12 times, so the total number of days spent with each classmate is:
12 days/classmate * 5 classmates = 60 days
She ate dinner with every pair of classmates 8 times, so the total number of days spent with each pair is:
8 days/pair * (5 choose 2) = 8 * 10 = 80 days
She ate dinner with every set of 3 classmates 10 times, so the total number of days spent with each set is:
10 days/set * (5 choose 3) = 10 * 10 = 100 days
She ate dinner with every foursome 5 times, so the total number of days spent with each foursome is:
5 days/foursome * (5 choose 4) = 5 * 5 = 25 days
She ate dinner with all five classmates 4 times, so the total number of days spent with the entire group is:
4 days * (5 choose 5) = 4 days
Now, let's calculate the total number of days she had dinner with at least one classmate:
60 days + 80 days + 100 days + 25 days + 4 days = 269 days
Finally, let's subtract this total from the total number of days during her stay:
70 days - 269 days = -199 days
Since the result is negative, it means Ayesha did not have dinner with none of her classmates on any day during her stay.