In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least one woman has to be necessarily selected?

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5 persons are to be selected out of 6 men and 4 women. At least, one woman has to be selected in all cases.

$Required number of ways =^{4} C_{1} \times^{6} C_{4} +^{4} C_{2} \times^{6} C_{3} +^{4} C_{3} \times^{6} C_{2} +^{4} C_{4} \times^{6} C_{1} = 60 + 120 + 60 + 6 = 246$

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A committee is to be formed comprising of 6 members such that there is a simple majority of men and at least 1 woman. The shortlist consists of 9 men and 6 women. In how many ways can this committee be formed?

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In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least one woman has to be necessarily selected?

5 persons are to be selected out of 6 men and 4 women. At least, one woman has to be selected in all cases.Required number of ways =4C1×6C4 + 4C2×6C3 + 4C3×6C2 +4C4×6C1= 60 + 120 + 60 + 6 = 246

5 persons are to be selected out of 6 men and 4 women. At least, one woman has to be selected in all cases.

$Required number of ways =^{4} C_{1} \times^{6} C_{4} +^{4} C_{2} \times^{6} C_{3} +^{4} C_{3} \times^{6} C_{2} +^{4} C_{4} \times^{6} C_{1} = 60 + 120 + 60 + 6 = 246$