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 Ataloss March 22nd, 2012 12:12 PM

Problem, is this right?

Can someone please help me with this problem? For part A I got 9! = 362 880. Am I on the right track?

Suppose that there are a total of 16 students in an elementary school class. The teacher assigns each of the students a report on a mainland country in either North America or South America (not including the United States) and each student is assigned a different country. The North American countries that the teacher has to pick from are Belize, Canada, Costa Rica, El Salvador, Guatemala, Honduras, Mexico, Nicaragua, and Panama (for a total of 9 countries) and the South American countries they have to pick from are Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, French Guiana, Guyana, Paraguay, Peru, Suriname, Uruguay, and Venezuela (for a total of 13 countries). Assuming that the order in which the countries are assigned doesn't matter:
a) In how many ways can the countries be assigned so that all of the North American countries are
assigned?
b) In how many ways can the countries be assigned so that there is at least one North American that is not assigned and at least one South American country that is not assigned?

 Ataloss March 22nd, 2012 09:16 PM

Re: Problem, is this right?

anyone?

 Isbell March 22nd, 2012 10:43 PM

Re: Problem, is this right?

Quote:
 Originally Posted by Ataloss (Post 109459) Can someone please help me with this problem? For part A I got 9! = 362 880. Am I on the right track? Suppose that there are a total of 16 students in an elementary school class. The teacher assigns each of the students a report on a mainland country in either North America or South America (not including the United States) and each student is assigned a different country. The North American countries that the teacher has to pick from are Belize, Canada, Costa Rica, El Salvador, Guatemala, Honduras, Mexico, Nicaragua, and Panama (for a total of 9 countries) and the South American countries they have to pick from are Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, French Guiana, Guyana, Paraguay, Peru, Suriname, Uruguay, and Venezuela (for a total of 13 countries). Assuming that the order in which the countries are assigned doesn't matter: a) In how many ways can the countries be assigned so that all of the North American countries are assigned? b) In how many ways can the countries be assigned so that there is at least one North American that is not assigned and at least one South American country that is not assigned?

We've a task, and the task required us to assign each student (there is a total of 16 students) a different country from a total of 9 North American countries and 13 South American countries and what you did is you only considered the number of ways to assign all 9 North American countries to any 9 students from 16 students.
What about the remaining 7 students? You still want to assign a country to each and everyone of them, do you?

Therefore,
$n=16P9\times13P7=3.59\times 10^{16}$
i.e. we first arrange any 9 students from 16 students so that each of them gets a country from North America, AND then we arrange any 7 South American countries out of 13 of them so that each country belongs to the remaining 7 students.

b.
In how many ways can the countries be assigned so that there is at least one North American that is not assigned and at least one South American country that is not assigned?
This one is a little bit challenging but a clear mindset will help.
We've a total of 9+13=22 countries and 16 students need to get a country. This means there is a (22-16=6) countries that will be left unselected.
The challenge now, is to carefully dealing with 6 countries so that the condition where 'at least one North American that is not assigned and at least one South American country that is not assigned' is satisfied.

OK, there are 5 cases of them.
Let the numbering below represents the countries that are left unselected by the students, NA represents the countries from North America and SA represents the countries from South America.
1) 1-NA, 5-SA
2) 2-NA, 4-SA
3) 3-NA, 3-SA
4) 4-NA, 2-SA
5) 5-NA, 1-SA

$n_1=9C1\times13C5\times16!=2.42\times 10^{17}$
I find we need to select any one of the NA country from 9 of them and any 5 SA countries from 13 of them so that we can put them away. This will ensure the condition that 'at least one North American that is not assigned and at least one South American country that is not assigned' is met.
Now, all that is left is the perfect 16 countries for 16 students! To arrange all these 16 students to get a country for each is simply 16!

By the same argument, we find

$n_2=9C2\times13C4\times16!=5.39\times 10^{17}$

$n_3=9C3\times13C3\times16!=5.02\times 10^{17}$

$n_4=9C4\times13C2\times16!=2.06\times 10^{17}$

$n_5=9C5\times13C1\times16!=3.43\times 10^{16}$

Therefore,
$n(total)=n_1+n_2+n_3+n_4+n_5=1.52\times10^{18}$

(P.S. Actually there is an easy way to solve for part b but I think it will be an advantage for you to learn from the basic.)

Quote:
 Originally Posted by Ataloss anyone?
If you don't get an answer immediately, just wait.
You need to be patient because we all do have real lives beyond helping you, OK?
Finally, please don't take this message amiss, it is not meant to offend!

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