Question

From $6$ different novels and 3 different dictionaries, 4 novels and 1 dictionaries are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangements is
A.less than 500
B.at least 500 but less than 750
C.at least 750 but less than 1000
D.at least 1000

Answer 1

Here, we will find the number of required arrangements by using the combination concept. A combination is a mathematical technique or a method of selecting an element from a set of elements in such a way that the order or sequence of selection does not matter.

Complete step-by-step answer:
We are given with 6 different novels and 3 different dictionaries. From these, we have to select 4 novels and 1 dictionary and arrange it in a row on a shelf so that the dictionary is always in the middle.
Number of possible ways of selecting 4 novels from $6$ different novels $= {}^6{C_4}$
Number of possible ways of selecting 1 dictionary from 3 different dictionaries $= {}^3{C_1}$
Since the dictionary should be placed at the middle, so the position of dictionary is fixed at the middle.
Arrangements of novel can be done in 4 ways $= 4!$
The total number of ways that 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle $= {}^6{C_4} \cdot {}^3{C_1} \cdot 4!$
Therefore, computing the combination, we get
$\Rightarrow$Required number of arrangement $= \dfrac{{6 \times 5 \times 4 \times 3}}{{1 \times 2 \times 3 \times 4}} \cdot \dfrac{3}{1} \cdot 4 \cdot 3 \cdot 2 \cdot 1$
$\Rightarrow$Required number of arrangement $= 15 \cdot 3 \cdot 4 \cdot 3 \cdot 2 \cdot 1$
Multiplying the terms, we get
$\Rightarrow$Required number of arrangement$= 1080$
The required number of arrangements is at least 1000.
Therefore, the total number of ways that 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle is at least 1000.
Hence, option (D) is the correct option.

Note: We have to clear about using the concept of Permutation and Combinations. The difference between combinations and permutations is order or the sequence of the elements. In permutations, the order of the elements matters, whereas in combinations the order or the sequence does not matter. A permutation is a method of arranging the elements in such a way that the sequence or linear order of arrangement matters, or if the set is already in order, then it is defined as rearrangement of its elements.