Discussion
A trekking group is to be formed having 6 members. They are to be selected from 3 girls, 4 boys and 5 teachers. In how many ways can the group be formed so that there are 3 teachers and 3 boys or 2 girls and 4 teachers?
Answer: A
The two cases are mutually exclusive, so their counts add.
3 teachers and 3 boys:
\[ {}^{5}C_{3} \times {}^{4}C_{3} = 10 \times 4 = 40 \]
2 girls and 4 teachers:
\[ {}^{3}C_{2} \times {}^{5}C_{4} = 3 \times 5 = 15 \]
\[ 40 + 15 = 55 \]
Both cases give exactly six members, as required, and no group satisfies both conditions at once - so nothing is double-counted.
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