Here's the question:
"A congress will be attended by two representatives from Colombia, three from Chile, four from Argentina and five from Brazil. Each of the $14$ representatives prepared their own speech, but only $6$ will be drawn to speak. If the draw rule provides that each of the four countries must have at least one representative speaking, the number of different ways to compose the set of six speeches that will be heard at the congress, regardless of the order, is equal to how much?"
I've been working on this question for two weeks. My answer is $1450$ – but I solved it via brute force. I literally counted all the possibilities of the rule being broken and subtracted from the $3003$ possibilities to form groups of 6 people out of $14$.
I tried to look at the problem with fewer countries, and fewer choices. It helped me to count, but I couldn't decipher the pattern with which the problem changes when adding a new representative or a new country.
Can someone help me?
Best Answer
A nice way to do this is to use the principle of inclusion-exclusion. Take all $\binom{14}6$ ways to choose $6$ people, then for each country, subtract the assignments where that country has no chosen representatives. The result is $$ \binom {14}6 -\underbrace{\binom{12}6}_{\text{Columbia missing}} -\underbrace{\binom{11}6}_{\text{Chile missing}} -\underbrace{\binom{10}6}_{\text{Argentina missing}} -\underbrace{\binom{9}6}_{\text{Brazil missing}} $$ However, there is a problem. Arrangements with two missing countries will be subtracted twice in the above computation. To fix this, we must add these doubly-subtracted arrangements back in. To the above, we add $$ +\underbrace{\binom{9}{6}}_{\text{Columbia + Chile}} +\underbrace{\binom{8}{6}}_{\text{Columbia + Argentina}} +\underbrace{\binom{7}{6}}_{\text{Columbia + Brazil}} +\underbrace{\binom{7}{6}}_{\text{Chile + Argentina}} +\underbrace{\binom{6}{6}}_{\text{Chile + Brazil}} $$ At this point, we are done! We would not be done if there were any arrangements counted by $\binom{14}6$ with three countries missing, but that is not possible.