MHB Solving Combinations Problem with up to 5 Roles: 40 Employees

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In a factory with 40 employees, the problem involves selecting a union of 5 people for 5 different roles, allowing employees to hold multiple roles. Since each of the 5 positions can be filled by any of the 40 employees, the calculation is based on sampling with replacement. Therefore, the total number of combinations is calculated as 40 raised to the power of 5, resulting in 40^5 possible unions. This approach differs from standard combination calculations, as it accounts for the possibility of the same employee occupying multiple roles. The final answer is 40^5 combinations for the union of 5 roles.
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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...
 
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Yankel said:
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.
 
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