# Combinations and selections to form a committee of various nationality of size 6 and size 3

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!

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