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?

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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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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?

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 = 12C6×8C2 + 12C5×8C3 + 12C4×8C4 = 25872 + 44352 + 34650 = 104874