China Mathematical Competition

Question

Seven students are arranged to attend five sporting events. It is required that students A and B cannot attend the same event, every event is attended by at least one student, and each student must attend one and only one event. Then the number of the arrangement plans meeting the required condition is _______. (the answer should be given in numerical value)

Accepted Answer

There are two possible cases that satisfy the required conditions:

(1) there is an event attended by three students — this case has

C_{7}^{3} \cdot 5! - C_{5}^{1} \cdot 5! = 3600

plans;

(2) there are two events each attended by two students — this case has

\frac{1}{2} (C_{7}^{2} \cdot C_{5}^{2}) \cdot 5! - C_{5}^{2} \cdot 5! = 11400

plans.

Therefore, there are

3600 + 11400 = 15000

plans that meet the required condition. ☐