10 books are to be lined up on the shelf

  1. How many ways can the books be lined up.
  2. If 5 of the books are identical math books and 2 are identical Science books and 3 are identical English books, how many ways can they be lined up?

I understood part '1' of the question which is really easy, however, I don't understand what needs to be done in part '2'. How does the question change if some of the books are identical and what is the answer?

  • $\begingroup$ Imagine if every book was the same, i.e. you have no way of telling the difference between the books. Then there would be only one way of lining them up, since switching the positions of two books would make zero difference. This is the idea behind the difficulty of question b. $\endgroup$
    – Isaac Ren
    May 19, 2020 at 9:42
  • $\begingroup$ It's a difference between permutations when the order of the books matters and combinations when the order of books of a particular kind does not matter. $\endgroup$
    – Vasili
    May 19, 2020 at 9:45
  • $\begingroup$ Can you tell why dividing the result of part (a) by $5!\ 3! \ 2!$ will give the answer? $\endgroup$
    – Tavish
    May 19, 2020 at 9:49

1 Answer 1


Answer to part 1

1) If 10 books are identical, obviously there is only one way to line up them.

2) If 10 books are not identical, each book is different from the rest. then it's a full arrangement. The answer is $A^{10}_{10}$ = 10! = 3628800

Answer to part 2

Assume there are 10 slots and each slot refers to a book. You have to choose 5 slots out of 10 to put Math books, there are $C^{5}_{10}$ kinds of combination conditions. Then you have to choose 3 slots out of 5 to put English books, which is $C^{3}_{5}$ kinds of combination conditions. The rest slots are for Science books, there is only 1 condition. So the answer is $C^{5}_{10} * C^{3}_{5} * 1 = 2520$


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