A committee of 6 is chosen from 10 men and 7 women so as to contain at least 3 men and 2 women. In how many ways can this be done if two particular women refuse to serve on the same committee? a. 850 b. 8700 c. 7800 d. none of these

Explanation

`because` Total number of men = 10
and total number of women = 7
We have to form a committee containing atleast 3 men and 2 women.
Number of ways `= ""^(10)C_(3) + ""^(7)C_(3) + ""^(10)C_(4) xx ""^(7)C_(2)`
If two particular women to be always there.
`therefore` Number of ways `= ""^(10)C_(4) + ""^(5)C_(0) + ""^(10)C_(3) xx ""^(5)C_(1)`
Total number of committee when two particular women are never together
=Total - Together
`= (""^(10)C_(3) xx ""^(7)C_(3) + ""^(10)C_(4) xx ""^(7)C_(2)) - (""^(10)C_(4) xx ""^(5)C_(0) + ""^(10)C_(3) xx ""^(5)C_(1))`
`= (120 xx 35 + 210 xx 21) - (210 + 120 xx 5)`
`= 4200 + 4410 - (210 + 600)`
`= 8610 - 810 = 7800`