Discussion
A student council of 10 members consists of two students from engineering school, three students from science school and five students from arts school. The university administration selects three students from the council at random. What is the chance that out of the selected students, two belong to the same school and the third belongs to different school?
Answer: A
The probability is 79/120.
There are \(\binom{10}{3} = 120\) equally likely selections. Count the complements:
\[ \begin{aligned} \text{all different} &: 2 \times 3 \times 5 = 30 \\ \text{all same} &: \tbinom{3}{3} + \tbinom{5}{3} = 1 + 10 = 11 \\ \text{exactly two same} &: 120 - 30 - 11 = 79 \end{aligned} \]
Directly: two engineering and one other \(1 \times 8 = 8\); two science \(3 \times 7 = 21\); two arts \(10 \times 5 = 50\); total 79. 89/120 and 69/120 miscount one of the three cases, and 2/120 is far too small: the arts students alone give 50 favourable selections.
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