Ways to arrange books

All the books can be arranged in $(2+3+2)!=7!$ ways

There are $3$ branches, three units of books: $\{$History$\}$,$\{$Geography$\}$,$\{$Science$\}$- Arranging branches $=3!$ ways.

Arranging the books within the branches:

History: $2!$

Geography: $3!$

Science:$2!$

Total $=3!(2!\times3!\times2!)=144$ ways


How many ways can the books be arranged? As you said = $(2+3+2)! = 7!$

If the books of the same subject need to be arranged together you need to calculate de permutations for the groups and multiply them by the permutations within every category.

$3! (2! \times 3! \times 2!) = 144$ ways

Groups permutatios x (history permutations x geography permutations x science permutations)


If the books of the same subject must be placed together, there are in essence three "packs," and these can be ordered in just $3! = 6$ ways, where I assume that the order within a pack is irrelevant. If that order is not irrelevant, you then have $3!=6$ ways to arrange the packs, then within the associate packs you have $2!=2$, and $3!=6$ and $2!=2$ ways to order the books. Thus the total is $3! 2! 3! 2! = 144$ ways.

If the seven books are distinct, one can indeed order them in $7!$ ways.

Tags:

Combinatorics

Permutations

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