Question

What will be the number of permutations of 10 objects taken all at a time in which particular 4 never comes together.
(a) $10!\,\times \,4!$
(b) $10!-4$
(c) $\dfrac{7!.6!}{4!}$
(d) $10!-7!.4!$

Answer 1

Hint: To solve this problem first we will consider the permutation of total 10 different object irrespective of those 4 objects then after that we will assume those 4 objects to be together as a one object and then will find the total number of permutations in that case. We have to find the total number of cases when the 4 objects are not together so we will subtract the second case from the total number of permutations of 10 objects to get the desired answer.

Complete step-by-step answer:
We have 10 different objects and we have to find the total number of permutations when the particular 4 objects are not together.
So first we will find out the total number of permutations of 10 different objects. i.e., we get
Total number of ways of arranging 10 different objects = $10!$
Now we will find the total number of cases when we consider the 4 given particular objects together as the one product so total number of objects we have now is 7,
6 different objects and 1 object with 4 elements together,
So now we get total number of ways of arranging 7 objects as = $7!$
And also 4 objects are together so number of ways of arranging them = $4!$
Hence total number of cases when 4 particular objects are together, we get
=$7!\,\times \,4!$
Now to find the total number of cases in which the 4 particular elements are not together we will subtract the number of cases when they are together from the total number of cases that are possible that are $10!$
Hence we get total number of cases when the 4 particular elements are not together as,
$=10!-7!.4!$

So, the correct answer is “Option d”.

Note: To solve these kinds of problems you need to read the given question properly otherwise you may make several mistakes like you may only find the case in which the 4 elements are together and not the one with not together. So read the question carefully and try to solve it step by step. And also remember that number of permutations for n different objects is given by $n!$.