In how many ways can seven books be arranged on a shelf if...

In how many ways can seven books be arranged on a shelf if
a) the books are all different?
b) two of the books are identical?
c) the books are different and the mathematics book must be on an end?
d) the books are different and four particular books must be together?

Key Concepts

Permutation of Distinct Objects

This concept refers to the number of ways to arrange a collection where every object is unique. The arrangements are called permutations and the count is determined by the factorial of the number of objects, which captures the idea that every sequential choice reduces the remaining options.

Permutations with Identical Items

When some objects are identical, the total count of unique arrangements must be adjusted to avoid overcounting. This is done by dividing the total permutations of all objects by the factorial of the number of repeated items, thereby correcting for arrangements that are indistinguishable from each other.

Restricted Permutations

This concept involves calculating arrangements under certain conditions or restrictions—such as fixing specific positions for certain objects. The method typically involves first placing the constrained items and then arranging the remaining ones, ensuring all restrictions are honored in the counting process.

Block Method in Permutations

The block method is a strategy used for counting arrangements when a subset of objects must remain together. In this approach, the grouped items are treated as a single unit for the purpose of counting overall arrangements, and then internal arrangements of the block are calculated separately.

Transcript

00:01 So this question is asking us about how seven books can be ranged on the shelf in different scenarios.

00:07 So in the first part, they're asking us about what will happen if every single book was different.

00:12 So you can imagine we have seven slots to put our books in.

00:16 And once we put one book in, we can't put it in again.

00:19 So we can only use each book once.

00:22 So in the first slot, we have seven books to choose from.

00:26 In the second slot, we only have six books less.

00:28 We have to choose from those.

00:30 And in the third slot, we only have five books left, so we choose from those.

00:34 And we can continue in a decreasing order.

00:37 And by the time we reach the last slot, there's only one book to pick from.

00:41 So when we do this, we can see this is just equal to 7 factorial or 5 ,040 different ways.

00:48 Now in the second part, it's asking us what happens if two books are identical.

00:52 So again, we can imagine our seven slots, but let's pretend these two slots the books are identical.

00:58 So if we have, say, a book called a here and another book called a, the order of these two books doesn't matter.

01:07 So we already know that there are seven factorial ways to range books that are completely different.

01:13 But we have to divide out all the options where we're flipping these two identical books because that's still the same order.

01:21 So there are two factorial ways to arrange them because in the first book we can pick from the two identical ones.

01:27 And then in the second slot only one of them.

01:29 So this is equal to 2 factorial.

01:32 So basically we get 5 ,040 divided by a 2 or 2520 ways...

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