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A country is attempting to form a new government.  The country has a number of ethnic groups which are somewhat antagonistic to each other.

During the government organization, 20 representatives from the country's ethnic groups are assigned to 4 different committees. These committees are:

1. Defense (6 positions)
2. Finance (4 positions)
3. Communications (5 positions)
4. Environment (5 positions)

The first of these committees is considered to be especially powerful, and so is considered to be very desirable.

In order for the the allocation of representatives from different ethnic groups to these committees to be considered fair, the allocation is supposed to be completely random.

A dispute arises because all 4 members of a particular ethnic group were assigned to the Defense committee.  As a result, a mediation panel is convened to determine the probability of the observed allocation, and thereby to determine whether the process was fair.

Determine whether the process was fair, based on the probability of the observed selection occurring.

Here are the steps to accomplish this:

1. Determine the number of ways, N, that the 20 representatives can be divided into groups of the appropriate sizes, to fill all of the jobs.

   Hint: See "Permutations with Non-Distinct Elements" in the course slides.

2. Let A denote the event of interest -- that is, the observed allocation of representatives.

   Determine the number of ways that this observed allocation can occur, n_A. 

   Note that this may be calculated by first assuming that all 4 of the representatives from the "privileged" ethnic group have already been chosen, and then determining the number of ways in which the remaining representatives may be allocated to the remaining positions. That is, the number of ways 16 representatives may be allocated to committees of sizes 2, 4, 5, and 5 respectively.

3. Because we are assuming a random assignment of allocations, each of the above possibilities has an equal probability, so the probability of each of the above possibilities is 1/N (they are also the only possibilities).

   Thus, the total probability of event A occurring is n_A/N . Calculate that value.

   Does that value seem reasonable, or does it indicate that some shenanigans has been going on?