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: The concept of permutations and combinations is used in the above question. First of all we will find the number of ways for the selection of superintendent ship. There are a total three candidates who are applying for this post out of which only one is selected. Then we will find the ways of disposing appointments for teachership. Therefore, the total number of ways can be calculated by simply multiplying the two cases above.

Complete step-by-step answer:
Now from the question,
No. of ways superintendent ship is chosen ${}^3{C_{_1}}$
Number of ways of disposing appointments for teacher ship = ${}^{8 - 2}{C_2} \times 4! \times 4! \times {}^4{C_2}$
= ${}^6{C_2} \times 4! \times 4! \times 1$
Total number of ways = ${}^3{C_1} \times {}^6{C_2} \times 4! \times 4!$
= 25920
Hence the correct option is D.

Note: Permutation and combination are the ways to represent a group by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. Both concepts are very important in mathematics.
Permutation occurs in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered.
Combination refers to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used.
The two key formula for permutation and combination are;
${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
${}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
A permutation is used for the list of data (where the order of data matter) and the combination is used for a group of data (where the order of data doesn’t matter)