In how many ways can $12$ different books be arranged on a shelf so that three particular books are never together?

They did take away method as Total ways - always together

But I want to do it like this

first I will select $3$ particular books out of $12$ books then I will arrange those $3$ books in $10$ slots. But I am not getting my answer.

Please help


  • 3
    $\begingroup$ You don't get to select the three books. The three books are selected for you, and fixed. All you have to do is find how many legal ways there are to place them. $\endgroup$
    – Arthur
    Commented Jan 17, 2018 at 12:27
  • $\begingroup$ Can you find then number of ways $12$ books can be arranged? Can you find the number of ways $12$ books can be arranged in such a way that the $3$ particular books are together? Then subtract. $\endgroup$
    – drhab
    Commented Jan 17, 2018 at 12:31
  • 2
    $\begingroup$ Not sure the rules are clear. Can two of the three books be next to each other or not? $\endgroup$
    – lulu
    Commented Jan 17, 2018 at 12:31

3 Answers 3


"Never together" means: No two of the three particular books are allowed to be adjacent. If they would have meant "not all three together" they would have said so.

You can arrange the $9$ ordinary books in $9!$ ways, creating $10$ slots where one of the three particular books may be placed. There are $10\cdot9\cdot 8$ ways to choose different slots for these three books.

The total number of admissible arrangements therefore is $9!\cdot720=261\,273\,600$.

  • $\begingroup$ Sir No offense but answer is 126×10!. what i did....I calculated two case's arrangements..first when two books are together and one is apart and second case when none of three books is together....then I add both arrangements and got my answer $\endgroup$ Commented Jan 18, 2018 at 4:30
  • $\begingroup$ I am not a master of language but my feeling is opposite. I do think it means two are allowed to be together. Anyway, to each his own. $\endgroup$
    – King Tut
    Commented Jan 18, 2018 at 13:22

Arthur tells you, the books are already selected.

Also lulu points out, condition "three particular books are never together" is not very clear, I take it to mean only that all three cannot be together, but two together and one apart is fine.

Now going your way, subtract from total ways, the ways in which all three are together.

Total ways are $12!$ and ways in which three are together are $10!\cdot 3!$ so our answer should be $12! - 3! \cdot 10!$


*THE MISTAKES OF YOUR THINKING * At first, you don't need to select them, they are already selected. Secondary, why you trying to select 3 place from 10 slots, that can't counts the possibilities where 2 of them stay together.
*THE RIGHT WAY TO THINK * at first calculates the all possibilities as $12!$ Then disclude the possibilities , when they all stay together as $ 3!×10!$ Then disclude it from all possibilities . The right answer is $12! -(3!×10! )$

  • 1
    $\begingroup$ Your count may include cases where two of the three are together, which may not be what the original question intended $\endgroup$
    – Henry
    Commented Jan 17, 2018 at 16:17

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