Question #8d994

We first choose one man and one woman, and then the third person. This will have no influence as there is no functional difference between the committee members.

Man: #5# possibilities
Woman: #4# possibilities
Third person: #9-2=7# ways

Answer seems to be: #4*5*7=140# ways, BUT :
The third person is either a man or a woman, between which the order is not important, so we have to divide by #2#
Real answer: #140//2=70#

The (b)-question is ambiguous: does it mean: if they marry inside the group? Or if they marry at all?

(a) Consider the two separate possibilities:
#2# women are selected
and
#2# men are selected

If #2# women are selected, the selection of women can be done in
#C(4,2) = (4 xx 3)/2 = 6# ways
and the selection of men in
5 ways
The combined number of ways of selecting 3 people of whom 2 are women can be done in
#6 xx 5 = 30# ways.

Similarly if #2# men are selected, this can be done in
#C(5,2) xx 4 = 40# ways.

The two possibilities are mutually exclusive so the total number of ways of making the described selection is
#30 + 40 = 70# ways.

(b) If only one couple is married (I assume intra-group marriage)
for the situation when 2 women are selected, there are
#C(3,2) = 3# ways of selecting the 2 women that do not include the married woman and therefore #3# (the original 6 ways minus 3) ways in which the married woman was selected; these #3# ways eliminate 3 cases where the woman's husband was selected.
So the combined number of ways of selecting 3 people of whom 2 are women and the man is not married to either woman is
#30 - 3 = 27# ways

For the situation when 2 men are selected, there are
#C(4,2) = 6# ways of selecting the 2 men that do not include the married man and therefore #C(5,2) - C(4,2) = 4# ways that include the married man. For each way that includes the married man, one of the original ways is eliminated. Therefore the combined number of ways of selecting 3 people of whom 2 are men and the woman is not married to either man is
#40 - 4 = 36#

and the total number of ways of making the described selection (with no married couple selected) is
#27 + 36 = 63# ways.