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Distributing distinct objects into distinct groups with none empty or a capacity limit

This type of question, asked 4 times: 2021 Paper 2, 10 Apr 2023, 2 Apr 2026 S1, 8 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    82
    The students S1,S2,…,S10S_{1},S_{2},\ldots,S_{10} are to be divided into 3 groups A, B and CC such that each group has at least one student and the group CC has at most 3 students. Then the total number of possibilities of forming such groups is

  2. Question 2

    10 Apr 2023

    152
    Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is:

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  3. Question 3

    2 Apr 2026 S1

    56
    The number of seven-digit numbers, that can be formed by using the digits 1,2,3,51,2,3,5 and 7 such that each digit is used at least once, is ;

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  4. Question 4

    8 Apr 2026 S2

    59
    A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is

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