If there are n jobs and m machines, then there will be ......….sequence of doing jobs.

Options

A. mn
B. `m(n!)`
C. `n^m`
D. `(n!)^m`

Answer

To solve the problem of determining the number of sequences of doing jobs when there are \( n \) jobs and \( m \) machines, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have \( n \) jobs and \( m \) machines. We need to find out how many different sequences there are for completing these jobs across the machines.

**Hint**: Think about how each job can be assigned to any of the machines.

2. **Identify the Formula**: The total number of sequences of doing jobs can be calculated using the formula:
\[ \text{Total sequences} = (n!)^m \]
where \( n! \) (n factorial) represents the number of ways to arrange \( n \) jobs, and raising it to the power of \( m \) accounts for the \( m \) machines.

**Hint**: Recall that \( n! \) is the product of all positive integers up to \( n \).

3. **Substitute the Variables**: In our case, we replace \( n \) with the number of jobs and \( m \) with the number of machines:
\[ \text{Total sequences} = (n!)^m \]

**Hint**: Make sure to keep track of the factorial notation.

4. **Final Expression**: Therefore, the final expression for the total number of sequences of doing jobs when there are \( n \) jobs and \( m \) machines is:
\[ (n!)^m \]

**Hint**: This expression indicates that for each machine, the jobs can be arranged in \( n! \) different ways.

### Conclusion:
If there are \( n \) jobs and \( m \) machines, then there will be \( (n!)^m \) sequences of doing jobs.