Question

In all problems, as usual in this course, show how you got the answer, in particular which formulas, rules and definitions you used, and why. Just writing a number is not enough for the full mark.

1. Combinations and permutations

A datacenter has 10 servers and 4 routers, which they want to connect. Each server and each router can have any number of cables attached to it.

(a) How many cables would they need to connect each server to each router directly?

(b) Suppose they connect each server to only one router. How many different ways are there torun the connections, assuming that each router has at least one server connecting to it? Here, assume every server and every router has a different name, and you need to find out how many ways to connect names of servers with names of routers.

(c) They need to order 10 cables to connect servers to routers. The cables come in 1ft, 2ft and 4 ftlengths. How many different orders can be placed, ranging from all 1ft cables to all 4ft with all the intermediate combinations 1ft, 2ft and 4ft cables?

(d) Now they want to arrange all servers and routers on two racks, each of which has 7 shelves.They want to have two routers on each rack, and want routers to be not on the ends of the racks and not next to each other. How many ways are there to put router shelves into those two racks? (Here, having router shelves 2 and 4 on rack 1 and 3 and 6 on rack 2 is different from having router shelves 3 and 6 on rack 1, with 2 and 4 on rack 2).

Answer 1

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