Is It Permutation or Combination for Arranging Books with Specific Conditions?

[sebastianbravom]

Homework Statement

in how many different ways can 7 different books be arranged in a row if
a. 3 specified books must be together,
b. two specified boks mus occupy both ends

Homework Equations

i don't udnerstand wether it is a permutation or a combination.

The Attempt at a Solution

a. 4! x 7!
b. 7 x 5! x 6

 

[wildman]

no.

Try first this: how many permutations (that means unique orderings) of the 7 books can you have if we have 7 distinct books? If we tie three books together, how is that in comparision to a single book? In other words, can we treat it like a single book or not? Then how many books or book groups do we have to recombine? If we have two specified books on either end, how many books can we move around?

 

[HallsofIvy]

Homework Statement

in how many different ways can 7 different books be arranged in a row if
a. 3 specified books must be together,
b. two specified boks mus occupy both ends

Homework Equations

i don't udnerstand wether it is a permutation or a combination.

Then you need to learn the definition of "permutation"! A permutation always involves different orders or arrangements. Combination" involve grouping different things without regard for order. Now, does this involve putting the books in different orders?

The Attempt at a Solution

a. 4! x 7!
b. 7 x 5! x 6

a) Think of the three books that must be together as a single book. Now you have 5 books- there are 5! ways of doing that. But for each of those, there are 3! ways of interchanging those 3 books.

b) Remove the two books that are to be at the ends. That leaves 5 books. Choose an order for those 5 books. There are 5! ways to do that. Then you have to decide which book to put on the left end. There are 2 ways to do that. Once that is done, you don't have to decide which book to put on the right end- you only have 1 left.