From a class of 14 boys and 10 girls, 10 students are to be chosen for a
Question:

From a class of 14 boys and 10 girls, 10 students are to be chosen for a competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made? 

Solution:

2 girls who won the prize last year are surely to be taken. So, we have to make a selection of 8 students out of 14 boys and 8 girls, choosing at least 4 boys and at least 2 girls.

Thus, we may choose:

(4 boys, 4 girls) or ( 5 boys, 3 girls) or ( 6 boys, 2 girls)

Therefore, the required number of ways $=\left({ }^{14} \mathrm{C}_{4} \times{ }^{8} \mathrm{C} 4\right)+\left({ }^{14} \mathrm{C}_{5} \times{ }^{8} \mathrm{C}_{3}\right)+\left({ }^{14} \mathrm{C}_{6} \times{ }^{8} \mathrm{C}_{2}\right)$

 

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