There are,
. 7 Icelanders;
. 8 Americans;
. 6 Japanese;
and we have that,
- The Icelanders are all men;
- Among the Americans there are 4 women;
- Among the Japanese there are 4 women;
compute the following:
- a) in how many different ways can be formed a committee of 6 people?
- b) In how many different ways can be formed a committee of 3 people with an ambassador for each nationality?
- c) In how many different ways can be formed a committee of 3 people with an ambassador for each nationality and exactly 1 woman?
- d) In how many different ways can be formed a committee of 3 people with an ambassador for each nationality and at least 1 woman?
I have done the following:
point a)
$7+8+6=21$ people in total. From these 21 I go to choose 6 of them.
$$\binom{21}{6} = 54264$$
point b)
I choose 1 people for each nationality:
$$\binom{7}{1} \cdot \binom{8}{1} \cdot \binom{6}{1} = 336$$
point c)
I don't understand totally this point. It could be understood in two different way as: 3 people + 1 women, or, 3 people that comprehend 1 women.
the first one, I think is:
$$\binom{7}{1} \cdot \binom{4}{1} \cdot \binom{2}{1} \cdot \binom{8}{1}$$
the second one is:
$$\binom{7}{1} \cdot \binom{4}{1} \cdot \binom{2}{1} + \binom{7}{1} \cdot \binom{4}{1} \cdot \binom{4}{1} = 56 + 112 = 168$$
i.e.
from 7 Icelander I choose 1,
from 4 American women I choose 1,
and from 2 Japanese men I choose 1.
OR
from 7 Icelander I choose 1,
from 4 American men I choose 1,
and from 4 Japanese women I choose 1.
point d)
at least 1 woman means that we don't want the the choice of 0 women from 8.
So I from the following, I toggle that I don't want:
$$\binom{7}{1} \cdot \binom{8}{1} \cdot \binom{6}{1} - \binom{8}{0} = 336 - 1 = 335$$
but, I am not sure about all these results.
Can you give me any suggestions? Many thanks!