Solving Combinations Problem with up to 5 Roles: 40 Employees

[Yankel]

Hello,

How do I solve this problem:

In a factory there are 40 employees. A union of 5 people is being chosen.
How many combinations are they, if the union of 5 people contains 5 different roles, and an employee can have more than one role (up to 5) ?

It is like sampling with replacement, so it isn't just:

$${40 \choose 5}$$Thanks...

 

[mathmari]

Hello,

How do I solve this problem:

In a factory there are 40 employees. A union of 5 people is being chosen.
How many combinations are they, if the union of 5 people contains 5 different roles, and an employee can have more than one role (up to 5) ?

It is like sampling with replacement, so it isn't just:

$${40 \choose 5}$$Thanks...

It is not ${40 \choose 5}$..

There are $5$ positions and $40$ employees.

For the first position, there are $40$ possible employees.
For the second position, there are again $40$ possible employees, since the one that is chosen for the first position can also be chosen for the second position.
For the third position, there are again $40$ possible employees, for the same reason.
For the $4^{th}$ position, there are $40$ possible employees.
Fot the $5^{th}$ position, there are $40$ possible employees.

So, there are $\displaystyle{40^5}$ possible ways to create the union.