The planner in charge of raw materials operations at a well known beer-producing company needs to obtain 5 truckloads of fresh water. The ve trucks available can go to any of 11 sources of water (mainly dams and reservoirs). The basic question is how many possible ways this can be achieved. The 11 sources all have dierent kinds of minerals in the water, so it is always important how many truckloads come from which source, but there are dierent conditions as follows. Find the number of ways in each case. You may leave your answer expressed in terms of binomials, factorials, integers raised to integer powers, or products and quotients of these, unless otherwise requested.(Important: make sure you explain what formulae you use, and why.
(a) It does not matter which truck brings which kind of water, and any sources can be
used by any number of trucks. Express your answer as a single integer.
(b) It does not matter which truck brings which kind of water, but no source can be
used by more than one truck. In this case, give the answer as a single integer.
(c) The company needs to keep track of how far each truck goes, so two arrangements
are counted as dierent if they send dierent trucks to any given source. Still, no
source can be used by more than one truck.
(d) The conditions in (c) hold, but additionally the company needs to record the order
that the trucks return with the water. How many possible outcomes are there?
(e) The conditions in (a) hold, but additionally there is one of the sources that may only
be used at most twice due to lack of supply. (The other sources are unrestricted).
Express your answer as a single integer.
