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 should be included. In how many ways the selection can be made.

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Solution

No of ways $10$ students can be selected
$5 B, 5 G | | 6 B, 4 G | | 4 B, 6 G$
Case(i):- $5 B =^{12} C_{5}, 5 G =^{10} C_{3} + 1$
Total $=^{12} C_{5} (^{10} C_{3} + 1)$
Case(ii):- $6 B =^{12} C_{6}, 4 G =^{10} C_{2} + 1$
Total $=^{12} C_{6} (^{10} C_{2} + 1)$
Case(iii) :- $4 B =^{12} C_{4}, 6 G =^{10} C_{4} + 1$
Total $=^{12} C_{4} (^{10} C_{4} + 1)$
Total $=^{12} C_{5} (^{10} C_{3} + 1) +^{12} C_{5} (^{10} C_{3} + 1) +^{12} C_{5} (^{10} C_{3} + 1)$

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5

boys and

4

girls) in the class. In how many ways:
(i) One student (either girl or boy) can be selected to represent the class.
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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 should be included. In how many ways the selection can be made.

No of ways 10 students can be selected 5B,5G||6B,4G||4B,6GCase(i):- 5B=12C5,5G=10C3+1Total =12C5(10C3+1)Case(ii):- 6B=12C6,4G=10C2+1Total =12C6(10C2+1)Case(iii) :- 4B=12C4,6G=10C4+1Total =12C4(10C4+1)Total =12C5(10C3+1)+12C5(10C3+1)+12C5(10C3+1)

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 should be included. In how many ways the selection can be made.