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In how many way can a committee of 4 women and 5 men be chosen from 9 women and 7 men, if Mr. A refuse to serve on the committee if Ms. B is a member?

(a) 1608 

(b) 1860 

(c) 1680 

(d) 1806

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(d) 1806

The required committee can be chosen in the following ways: 

I. When Ms. B is a member and Mr. A refuses to serve. [Selection has to be made of 3 women from 8 women as Ms B is already there and 5 men from 6 men as Mr A is excluded] 

II. When Ms B is not a member and Mr. A can serve. (Selection is to made of 4 women from 8 women as Ms. B is excluded and 5 men from 7 men) 

∴ Total number of ways in which the committee can be formed. 

= 8C3 × 6C5 + 8C4 × 7C5 

= \(\frac{8\times7\times6}{3\times2}\times6\) + \(\frac{8\times7\times6\times5}{4\times3\times2}\) x \(\frac{7\times6}{2}\) 

= 56 × 6 + 70 × 21 = 336 + 1470 = 1806.

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