Answer to Question #226534 in Statistics and Probability for Pavi

Question

Suppose that a certain football tournament consists of 16 teams: 6 from Europe, 3 from South America, 3 from Africa, 2 from North America and 2 from Asia. Four teams will make the finals. For each of the following, show working to justify your answer. Evaluate all answers as numbers (i.e. do not leave them in terms of combination notation, or factorials). How many different possible combinations of finalists are there if

(a) there are no restrictions?

(b) exactly one South American team is in the finals?

(c) no team from North America makes the finals?

(d) at least two non-European teams make the finals?

(e) The four finalists are from different regions?

Answer 1

Solution:

(a):

There are total 16 teams from which 4 teams will make the finals.

Since, there are no restrictions.

Therefore, the number of ways 4 teams from 16 teams will make to the finals is the number of ways in which 4 teams can be selected from 16 teams which is $^{16}C_4=1820$

(b):

To ensure that exactly one South American team is in the finals, we first select 1 out of 3 South American teams. And then select the remaining 4−1=3 teams from the remaining 16−3=13

 teams where there is no restrictions.

Therefore, the total number of ways in which 1 of 3 South American teams can be selected and 3 out of 13 teams can be selected is $=^3C_1\times^{13}C_3=3\times 286=858$

(c):

To ensure that no North American team make the finals, we first remove 2 North American teams from the 16 teams and then select all 4 teams from those 16−2=14 teams where there is no restrictions.

Therefore, the total number of ways in which 4 teams can be selected from 14 teams is $=^{14}C_4=1001$

(d):

At least two non-European teams make the finals in any of the followings ways:

i) 2 non-European teams and 2 European teams make the final

ii) 3 non-European teams and 1 European teams make the final

iii) 4 non-European teams and no European teams make the final

Now, there are 6 European teams and 16−6=10 non-European teams

Therefore, the total number of ways in which at least 2 non-European teams make the finals is

$=^{10}C_2\times^6C_2+^{10}C_3\times^6C_1+^{10}C_4\times^6C_0 \\=45\times 15+120\times 6+210\times 1 \\=1605$

(e):

The four finalists will be from different regions if any of the following cases happen:

i) The four finalists are from Europe, South America, Africa and North America

ii) The four finalists are from Europe, South America, Africa and Asia

iii) The four finalists are from Europe, South America, North America and Asia

iv) The four finalists are from Europe, Africa, North America and Asia

v) The four finalists are from South America, Africa, North America and Asia

Then the total number of ways in which four finalists will be from different regions is

$=^6C_1\times ^3C_1\times ^3C_1\times ^2C_1+^6C_1\times ^3C_1\times ^3C_1\times ^2C_1 \\+^6C_1\times ^3C_1\times ^2C_1\times ^2C_1+^6C_1\times ^3C_1\times ^2C_1\times ^2C_1 \\+^3C_1\times ^3C_1\times ^2C_1\times ^2C_1 \\=108+108+72+72+36 \\=396$