First slide

Combinations

Question

The students $S_{1}, S_{2}, ..., S_{10}$ are to be divided into 3 group A,B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is_____________

Moderate

Solution

$\begin{array}{l} = {}^{10}C_{3} (2^{7} - 2) + {}^{10}C_{4} (2^{6} - 2) + \dots . . + {}^{10}C_{10} \\ = \sum_{r = 3}^{10} {}^{10}C_{r} 2^{10 - r} - 2 \sum_{r = 3}^{10} {}^{10}C_{r} \\ = [(1 + 2)^{10} - ({}^{10}C_{0} 2^{10} + {}^{10}C_{1} 2^{9} + {}^{10}C_{2} 2^{8})] \\ - 2 [2^{10} - ({}^{10}C_{0} + {}^{10}C_{4} + {}^{10}C_{2})] = 39449 \end{array}$

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