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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!

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Everything looks right except problem (d). Your strategy of taking the combinations from (b) and subtracting off the combinations with zero women is good. However, there are more than one combinations with zero women.

The number of choices with zero women is:

  1. one of the seven icelandic men
  2. one of the four american men
  3. one of the two japanese men

so ${7\choose 1}\cdot {4\choose 1} \cdot {2\choose1} = 56.$ So you get $336-56=280.$

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