# Question #8d994

Mar 19, 2015

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: $4 \cdot 5 \cdot 7 = 140$ 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?

Mar 19, 2015

(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
$C \left(4 , 2\right) = \frac{4 \times 3}{2} = 6$ 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
$6 \times 5 = 30$ ways.

Similarly if $2$ men are selected, this can be done in
$C \left(5 , 2\right) \times 4 = 40$ 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
$C \left(3 , 2\right) = 3$ 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
$C \left(4 , 2\right) = 6$ ways of selecting the 2 men that do not include the married man and therefore $C \left(5 , 2\right) - C \left(4 , 2\right) = 4$ 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.