From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls.
Question:
From a class of 12 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:
Two girls who won the prizes last year are to be included in every selection.
So, we have to select 8 students out of 12 boys and 8 girls, choosing at least 4 boys and 2 girls.
Number of ways in which it can be done $={ }^{12} C_{6} \times{ }^{8} C_{2}+{ }^{12} C_{5} \times{ }^{8} C_{3}+{ }^{12} C_{4} \times{ }^{8} C_{4}=25872+44352+34650=104874$
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