✒️[tex]\large{\mathcal{ANSWER}}[/tex]
[tex]======================[/tex]
- Jobs can be distributed in 36 different ways between companies.
Explanation:
This question involves a case of combinatorial analysis, where a permutation with repeated elements takes place. We have 4 jobs and only 3 companies that will be hired, so one of these companies will be hired twice. Let's look at the combination possibilities:
[tex]------------------[/tex]
Note that the order matters to us as it represents the jobs assigned to each company. Therefore, we can interchange the letters A, B and C so that the division of labor is different. In addition, there are 3 possibilities of permutation, as we can hire company A twice, or B or C.
[tex]------------------[/tex]
Also note that we have a repeating element in this permutation. For cases like this, we use the formula:
[tex] \: \boxed{p = \frac{n!}{a!}} [/tex]
Where N is the number of elements exchanged and A is the number of times any of them are repeated. Let's calculate:
[tex]p = \frac{4!}{2!} [/tex]
[tex]p = \frac{4 \times 3 \times 2!}{2!} [/tex]
[tex]p = \frac{4 \times 3 \times \cancel2!}{\cancel2!} [/tex]
[tex]p=12[/tex]
> Remember: we have 3 options of companies that can be hired twice, so we multiply by 3:
[tex]p = 12 \times 3[/tex]
[tex] \: \boxed{p = 36 \: way} [/tex]
[tex]======================[/tex]
