Question

An English school and a Vernacular school are both under one superintendent. Suppose that the superintendent ship, the four teacher ship of English and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendent ship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is
(a) 4320
(b) 268
(c) 1080
(d) 25920

Answer 1

Hint: According to the question there are a total of 9 vacancies, 1 superintendent ship, 4 teacher ship in English school and four teacher ship in Vernacular school. The total number of applicants are 11, out of which 3 are exclusively for superintendent ship, so there are 8 candidates for teacher ship and 8 vacancies as well. So, the thing is all 8 candidates for teacher ship will get the job while 1 out of the 3 will be selected, i.e., $^{3}{{C}_{1}}$ to get the job. Out of the 8 teacher ship candidates, 2 are exclusively for English school, so they are fixed. So, select 2 out of the six left candidates who will be there in the English school and rest 4 will automatically be appointed in the vernacular school.

Complete step by step solution:
Let us start the solution to the above question by interpreting the things given in the question.
According to the question there are a total of 9 vacancies, 1 superintendent ship, 4 teacher ship in English school and four teacher ship in Vernacular school. The total number of applicants are 11, out of which 3 are exclusively for superintendent ship, so there are 8 candidates for teacher ship and 8 vacancies as well. So, the thing is all 8 candidates for teacher ship will get the job while 1 out of the 3 will be selected, i.e., $^{3}{{C}_{1}}$ to get the job.
Now we will move to fill the teaching vacancies. Out of the 8 teacher ship candidates, 2 are exclusively for English school, so it is fixed that they would get a job at the English school and the way of doing so is 1.
Now we are having 6 teacher ship candidates left and 6 vacancies, 2 in English school and 4 in vernacular school. So, first let us select 2 out of 6 candidates who will teach in English school and the left out will automatically get the teaching job in the vernacular school. So, the number of ways of selecting 2 out of 6 candidates is $^{6}{{C}_{2}}$ and to get the number of ways of selecting teachers in both types of schools together we need to multiply two 4! To this, as in each school the 4 vacancies can be interchanged among them. So, the number of ways of selecting teachers is $^{6}{{C}_{2}}\times 4!\times 4!$
Now as we have to select all of them together, we will multiply the ways of selecting individually. So, the answer is: $^{3}{{C}_{1}}{{\times }^{6}}{{C}_{2}}\times 4!\times 4!=\dfrac{3\times 6\times 5}{2}\times 24\times 24=25920$
Hence, the answer to the above question is option (d).

Note: The most important thing to understand is the 4 teaching posts in each type of schools are not similar and the teachers can be given one of the 4 jobs which is why we multiply two 4! to compensate for the possible arrangements in each school. Also, remember the question is not suggesting that the teacher ship candidates will not get the superintendent job, but the numbers suggest it, because if one of the teacher ship candidates gets the superintendent ship job, the number of teaching vacancies will be more than the candidates for it.