There are 6 jobs with distinct difficulty levels, and 3 computers with…
2021
There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that:
- The fastest computer gets the toughest job and the slowest computer gets the easiest job.
- Every computer gets at least one job.
The number of ways in which this can be done is ___________.
Attempted by 102 students.
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Correct answer: 65
Answer: 65
The assignment of the toughest job to the fastest computer and the easiest job to the slowest computer is fixed, so remove those two jobs from consideration.
That leaves 4 remaining distinct jobs to assign to the three computers. Each of these 4 jobs can be assigned independently to any of the three computers, giving 3^4 total assignments for the remaining jobs.
We must ensure every computer gets at least one job. Since the fastest and slowest already have one job each, the only extra requirement is that the middle-speed computer gets at least one of the remaining 4 jobs.
Count assignments where the middle computer gets none: then each of the 4 jobs goes to either the fastest or the slowest computer, giving 2^4 such assignments.
Valid assignments = total assignments − assignments with middle receiving none = 3^4 − 2^4 = 81 − 16 = 65.
Therefore, the number of ways to assign the jobs under the given constraints is 65.