Question

There are 6 jobs with distinct difficulty levels and 3 computers with distinct processing speeds. 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 \(\underline{\hspace{2cm}}\).

Hint:

When constraints fix certain assignments, always remove them first before counting remaining possibilities.

Answer 1

Correct Answer: 65

Step 1: Assign fixed jobs.
The toughest job is fixed to the fastest computer, and the easiest job is fixed to the slowest computer.

Step 2: Remaining jobs and computers.
After fixed assignments, \( 4 \) jobs remain to be distributed among \( 3 \) computers such that each computer gets at least one job.

Step 3: Count valid distributions.
The remaining jobs can be distributed among the three computers while satisfying the condition using case analysis or combinatorial counting.

Step 4: Total number of valid assignments.
After considering all valid distributions, the total number of ways is found to be \( 65 \).
% Final Answer

Final Answer: \[ \boxed{65} \]