Seven women and 9 men are on the faculty of a math departmen

Seven women and 9 men are on the faculty of a math department. How many ways are there to select a committee of 5 members of the department if at least one woman must be on the committee? What if at least one woman and one man must be on the committee?

Solution

Given: 7 women and 9 men are on the faculty of a math department.

then

    select a committee of 5 members of the department if at least one woman must be on the committee is:

              ncr(16,5) - ncr(9,5)= 4242

                            7C1*9C4 + 7C2*9C3 + 7C3*9C2 + 7C4*9C1 + 7C5*9C0 = 4242

And at least one woman and one man must be on the committee is:

                       ncr(16,5) - ncr(9,5)-ncr(7,5)= 4221

                                                                             4242 - 7C5*9C0 = 4221