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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?
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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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