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China Mathematical Competition

Combinatorics Difficulty 4.6 AIME Find the answer China

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)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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. ☐

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