In how many ways can 7 different objects be divided among 3 persons so that…
2025
In how many ways can 7 different objects be divided among 3 persons so that either one or two of them do not get any object?
- A.
381
- B.
180
- C.
36
- D.
84
Attempted by 1929 students.
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Correct answer: A
Key idea: Count distributions where either two persons get nothing (all objects to one person) or exactly one person gets nothing (two persons share the objects with each receiving at least one).
Case 1: Two persons get nothing. All 7 distinct objects go to a single person, giving 3 ways (choose which person).
Case 2: Exactly one person gets nothing. Choose the two people who receive objects in 3 ways. For a fixed pair, each of the 7 distinct objects can go to either of the two people, giving 2^7 = 128 assignments; exclude the 2 assignments where one of the two receives all objects, so there are 2^7 - 2 = 126 valid assignments per pair. Thus this case yields 3 * 126 = 378 ways.
Total number of ways = 3 (Case 1) + 378 (Case 2) = 381.
Answer: 381