How many ways can the books be lined up?
10 books are to be lined up on the shelf
- How many ways can the books be lined up.
- 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?
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$\begingroup$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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