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let us consider this problem:

Chelsea has a bookshelf consisting of ten classics: four Russian novels, three British novels, two French novels, and a German novel. If she wants to make sure that the novels are always grouped according to country, how many ways can she arrange the novels?

my attempt is following because these novels should be arranged according to country,it means that i should multiply number of arrangement of Russian novels together by number of British novels arrangement by French and one German Novels number of arrangement,which means that


but in answer there is $24*288=6912$ where $24$ comes from?does it means that there is $24$ ways first book i could choose? $4*3*2*1=24$?

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up vote 2 down vote accepted

Consider the big group. So you have Russian, British, French, and German. You can arrange them $4!$ ways. For each type of book, you can arrange them $4!$, $3!$, $2!$, and $1!$ ways (which is what is in your attempt.)

So the final answer is $4!4!3!2!1!=6912$.

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it means that $4!$ is related to number of country,not number of books right? – dato datuashvili Aug 1 '13 at 6:01
and also it means that order is important right? – dato datuashvili Aug 1 '13 at 6:11
yes, $4!$ refers to the country, and the order matters. – vantonio1992 Aug 1 '13 at 12:02

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