Question

In: Statistics and Probability

The worker in charge of raw materials operations needs to obtain 5 truckloads of fresh water....

The worker in charge of raw materials operations needs to obtain 5 truckloads of fresh water. The five 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 different kinds of minerals in the water, so it is always important how many truckloads come
from which source, but there are different conditions as follows. Find the number of ways in each case.
1) 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.
2) 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.
3) The company needs to keep track of how far each truck goes, so two arrangements
are counted as different if they send different trucks to any given source. Still,
no source can be used by more than one truck.
4) The conditions in (3) hold, but additionally the company needs to record the
order that the trucks return with the water. How many possible outcomes
are there?
5) The conditions in (1) 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.

Solutions

Expert Solution

1)

If it does not matter which truck brings which kind of water, each truck can be send to a given source in 11 ways.

number of ways = = 48828125 ways

2)

If it does not matter which truck brings which kind of water, but no source can be used by more than one truck, then the order in which trucks are assigned is not relevent.

number of ways =

= (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1)

= 462 ways

3)

If it does not matter which truck brings which kind of water, but no source can be used by more than one truck, then the order in which trucks are assigned is relevent.

number of ways =

= 11 * 10 * 9 * 8 * 7

= 55440 ways

4)

Number of ways the trucks return with the water =

= 55440 ways

Total number of ways if (3) also holds = 55440 * 55440 = 3073593600 ways

5)

If one of the sources that may only be used at most twice due to lack of supply, then that source can be used by at most 2 trucks.

If it is used by no trucks, number of ways,other 5 trucks are assigned to remaining 10 source = = 9765625 ways

If it is used by a truck, number of ways, other 4 trucks are assigned to remaining 10 source = = 1048576 ways

If it is used by two trucks, number of ways, other 3 trucks are assigned to remaining 10 source = = 59049 ways

Total number of ways = 9765625 + 1048576 + 59049 = 10873250 ways

orchestra answered 2 years ago
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