# Question about possibilities of a team

consider an experiment that consists of determining the type of job - either blue-collar or white collar- and the political affiliation -republicans, democratic or independent - of the 15 members of an adult soccer team. how many outcomes are in the sample space?

I calculated it by doing the following calculation:

15x2x3=90 outcomes

However the answer is 6^15. How did they calculate it like this and why is my calculation wrong?

• Could you explain how you got to $15 \times 2 \times 3$? Jun 22 at 2:57
• Yes, I had 15 people, and for each 15 I had two possibilities for their occupations being either blue collar or white collar, and then three possibilities for their political affiliations. So I multiplied the possibilities and came up with 15x2x3. Jun 22 at 3:11

for every player, the total number of pairings (job,political affiliation) are $$\binom{2}{1}\times \binom{3}{1} = 6$$. Assuming that pairings of any 2 soccer players are independent, each player has 6 options of the pairings. thus it would be $$\underbrace{6 \times 6 \times 6 \cdots 6}_{\textrm{15 times}} = 6^{15}$$.

• Thanks, I understand now how they arrived at this calculation. However, I am still not sure why the calculation I derived was incorrect? Jun 22 at 3:12

Using the basic counting principle of multiplication, it must be $$2^{15}\times 3^{15}=6^{15}$$.

• Thanks this makes it clear how it was solved. Jun 23 at 14:04

$$\underbrace{6 \times 6 \times 6 \cdots 6}_{\textrm{15 times}} = 6^{15}$$

This is because of the fundamental principle of product that states that if a job is doable in $$m$$ ways and another job is doable in $$n$$ ways, the number of ways in which both the jobs are doable is $$m\times n$$.

Since there are $$6$$ choices in each of the $$15$$ cases, it is $$6^{15}$$.

Yes, it is an application of the basic principle of counting.

$$2^{15}\times 3^{15}=6^{15}$$