# Counting Problem - discrete math. There are three computers A, B, and C. Computer A has 10 tasks, Computer B has 15 tasks, and Computer C has 20 tasks. Each computer must complete its own tasks in order. After, each computer sends its output to a shared fourth computer. How many different orders can the outputs arrive at the fourth computer.

There are three computers A, B, and C. Computer A has 10 tasks, Computer B has 15 tasks, and Computer C has 20 tasks. Each computer must complete its own tasks in order. After, each computer sends its output to a shared fourth computer. How many different orders can the outputs arrive at the fourth computer.
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Step 1
This is not a permutation problem, because the order of tasks for each of the three computers is fixed; the only thing that varies is how the tasks for the three computers are interleaved. Once we know which 10 of the $10+15+20=45$ positions in the output are occupied by A’s tasks, we know which of A’s tasks is in each of those 10 positions: they must have been done in order. Similarly, once we know which 15 positions have the output of B’s tasks, we know which of B’s tasks is in each of those positions.
Step 2
How many ways are there to choose the 10 positions in the output for A’s tasks?
Once that’s been done, how many ways are there to choose 15 of the remaining positions for B’s tasks?
At that point all 20 of the positions that still remain must be filled with C’s tasks in their proper order, so there are no more choices to be made. Putting the pieces together, how many different orders are there in which the 45 outputs can arrive at the fourth computer?