How many ways can 5 different books be arranged on a shelf if I there are no restrictions II 2 books are always together?
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I have $5$ books $A,B,C,D,E $ and spots as $$\underline{A} \ \underline{B} \ \underline{C} \ \underline{D} \ \underline{E}$$ These can be arranged in $5!$ ways. So suppose that $A,B$ must remain together. Can I treat $AB$ as a single element and consider $$\underline{AB} \ \underline{C} \ \underline{D} \ \underline{E}$$ even though this has $5$ books but $4$ spots? These can be arranged as $$\underline{AB} \ \underline{C} \ \underline{D} \ \underline{E} \\ \underline{C} \ \underline{AB} \ \underline{D} \ \underline{E} \\ \vdots \\ \underline{C} \ \underline{D} \ \underline{E} \ \underline{AB} $$ so I have $4$ different arrangements that can be ordered in $4!$ ways so the total would be $5! - 4\cdot(4!)= 24$?
asked Nov 25, 2021 at 9:53
$\endgroup$ 3 $\begingroup$ As mentioned in the comments, you must also consider the order of the two books that remain together. Place the two books that must remain together in a box. Then we have four objects to arrange, the box and the other three books. The four objects can be arranged in $4!$ orders. The two books within the box can be arranged in $2!$ orders. Hence, the number of possible arrangements of five books if two of the books must remain together is $4!2!$. $\endgroup$ Here are the five books: Let's use slots like we did with
the license plates: We'll fill each slot -- one at a time... Then we can use the counting principle! The first slot: We have all 5 books to choose from to fill this slot. Let's say we put book C there... Now, we only have 4 books that can go here...  How many books are left for this slot? See it? Whoa, dude! That's 5! So, there are 120 ways to arrange five books on a bookshelf. Was the answer to our 3-book problem really 3! ? Yep! Will this always work? TRY IT: How many ways can eight books be arranged on a bookshelf? (reason it out with slots)
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How many ways can five distinct books be arranged in two bookshelves?Therefore, the total number of ways in which we can arrange 5 distinct books in 2 shelves is = 6 x 120 = 720.
How many ways can five books be ordered from left to right on a shelf?giving 5×4×3×2×1=120 choices for arranging the books (from left to right).
How many ways can you arrange 8 books on a shelf if only 5 books can fit at a time?8 books can be placed on a shelf in 8*7*6*5*4*3*2*1 ways. That equals to 40320. It can also be written as 8!
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