Question

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on shelfs of which the dictionary is always in the middle. The number of such arrangement is
A. At least 500 but less than 750
B. At least 750 but less than 1000
C. At least 1000
D. Less than 500

Answer 1

In this question we should know the formula for permutation and combination and the condition in which permutation and combination should be used. If we have to select things from a number of things then we use combinations, and if we have to arrange things then we use permutation
Formula for combination is
$\mathop n\nolimits_{\mathop c\nolimits_r } = \dfrac{{n!}}{{(n - r)!r!}}$
Formula for permutation is =
$\mathop n\nolimits_{\mathop P\nolimits_r } = \dfrac{{n!}}{{(n - r)!}}$

Complete step-by-step solution:
Let $\mathop N\nolimits_1 ,\mathop N\nolimits_2 ,\mathop N\nolimits_3 ,\mathop N\nolimits_4 ,\mathop N\nolimits_5 ,\mathop N\nolimits_6$ be 6 novels and $\mathop D\nolimits_1 ,\mathop D\nolimits_2 ,\mathop D\nolimits_3$ be 3 dictionaries.
Now first of all we have to select 4 novels out of 6 novels, for this we have to use combination,
Therefore number of ways to select 4 novels $\mathop { \Rightarrow 6}\nolimits_{\mathop c\nolimits_4 } = \dfrac{{6!}}{{(6 - 4)!4!}}$ $= 15$
Now we have to select 1 dictionary out of 3 dictionaries, for this we again have to use combination,
Therefore no. of ways to select1dictionary= $\mathop 3\nolimits_{\mathop c\nolimits_1 } = \dfrac{{3!}}{{(3 - 1)!1!}} = 3$
After selecting 4 novels and 1 dictionary, we have to arrange them in a row so that the dictionary is always in the middle, as we have to arrange them so we have to use permutation.
For this arrangement, we have to fix the position of the dictionary in the middle. The arrangement is looking like that
$N,N,D,N,N$
Where N stands for novel and D stands for dictionary
Now we have to arrange the 4 novels without changing the position of dictionary Therefore, the
the number of such arrangements is.
$\mathop 4\nolimits_{\mathop P\nolimits_4 } = \dfrac{{4!}}{{(4 - 4)!}} = 24$
Now, no. of ways to select the 4 novels and 1 dictionary and arranging them in a row on a shelf is
$\mathop 6\nolimits_{\mathop c\nolimits_4 } ,\mathop 3\nolimits_{\mathop c\nolimits_1 } ,\mathop 4\nolimits_{\mathop P\nolimits_4 } = 15 \times 3 \times 24 = 1080$

Therefore, the correct answer is option C.

Note: Many students get confused about which positions they should consider to arrange things. So, we have to take care that we have to arrange those things which are not fixed in its position .And don’t consider that position to arrange things which are fixed.