In how many ways can a committee of 3 ladies and 4 gentlemen be select

1.	In how many ways can a committee of 3 ladies and 4 gentlemen be selected from agathering of 8 ladies and 7 gentlemen? What will be the number of ways in case of Ms Arefusing to be in the committee if Mr. B is a member?
Answer» 3 ladies out of 8 can be selected in ⁸C₃ ways and 4 gentlemen out of 7 in ⁷C₄ ways. Now each way of selecting 3 ladies is associated with each way of selecting 4 gentlemen. Hence, the required number of ways = ⁸C₃ × ⁷C₄ =56×35=1960. We now find the number of committee of 3 ladies and 4 gentlemen in which both Mrs A and Mr B are member. In this case we can select 2 other ladies from the remaining 7 in ⁷C₂ ways and 3 other gentlemen from the remaining 6 in ⁶C₃ ways. ∴ The number of ways in which both Mrs A and Mr B are always included = ⁷C₂ × ⁸C₃ =21×20=420 Hence the required number of committee in which Mrs A and Mr B do not serve together =1960=420=1540.

In how many ways can a committee of 3 ladies and 4 gentlemen be selected from agathering of 8 ladies and 7 gentlemen? What will be the number of ways in case of Ms Arefusing to be in the committee if Mr. B is a member?

Answer»

3 ladies out of 8 can be selected in ⁸C₃ ways and 4 gentlemen out of 7 in ⁷C₄ ways.

Now each way of selecting 3 ladies is associated with each way of selecting 4 gentlemen.

Hence, the required number of ways = ⁸C₃ × ⁷C₄ =56×35=1960.

We now find the number of committee of 3 ladies and 4 gentlemen in which both Mrs A and Mr B are member.

In this case we can select 2 other ladies from the remaining 7 in ⁷C₂ ways and 3 other gentlemen from the remaining 6 in ⁶C₃ ways.

∴ The number of ways in which both Mrs A and Mr B are always included = ⁷C₂ × ⁸C₃ =21×20=420

Hence the required number of committee in which Mrs A and Mr B do not serve together =1960=420=1540.

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