15. A teacher asked each student of a class to write down the number of classmates with the same surname as his, as well as the number of classmates whose birthday is the same as his. Among the answers received, each of the numbers $0,1,2, \ldots, 10$ has appeared. Find the minimum number of students in the class.
老師要求某班的每位學生分別寫下班中姓氏和自己相同的同學數目, 及班中和自己同一天生日的同學數目。在收到的答案中，0、1、2、..、10 各數均有出現。求該班學生人數的最小可能值。
15. 33
15. Divide the students into groups according to their surname. If there are $k$ students in a group, then every student in the group will write $k-1$ for the question on surnames. The same is true if we divide the students into groups according to birthdays.
Since each of $0,1,2, \ldots, 10$ has appeared as an answer, there is at least one group of each of the sizes $1,2,3, \ldots, 11$. As each student belongs to two groups (one 'surname group' and one 'birthday group'), the number of students must be at least $(1+2+3+\cdots+11) \div 2=33$.
Finally, it is possible that there are 33 students when, for instance, the answers to the 'surname question' are all $2,8,9,10$ (i.e. there are four 'surname groups' with $3,9,10,11$ students) and the answers to the 'birthday question' are all $0,1,3,4,5,6,7$ (i.e. there are seven 'birthday groups' with $1,2,4,5,6,7,8$ students). Hence the answer is 33 .