Question #15385

There are 7 English, 5 Marathi and 4 Hindi Books. In how many ways can they be arranged in a shelf such that,
(i) the books are language wise.
(ii) the books are not language wise strictly, although some but not all books of same language may be together.

Expert's answer

i) First choose the position of the language , this can be done in 3! = 6 ways, next find the number of all permutations of the books inside their language, that can be done by 7! ways for English, 5! for Marathi and 4! for Hindi. By the combinatorial rule of product this arrangement can be done in 3!4!5!7! ways.

ii) All the books can be arranged in (7 + 5 + 4)! = 16! ways. Find the number of ways that do not satisfy our arrangement condition (choose the position of the ‘first book‘ in English or Marathi or Hindi, permutate all the books in English, Marathi or Hindi, but we count twice the number of ways, when all the books are arranged language wise) 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! − 7!4!5!. Hence, the answer is 16! − 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! + 7!4!5!.

Answer. 16! − 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! + 7!4!5!.

ii) All the books can be arranged in (7 + 5 + 4)! = 16! ways. Find the number of ways that do not satisfy our arrangement condition (choose the position of the ‘first book‘ in English or Marathi or Hindi, permutate all the books in English, Marathi or Hindi, but we count twice the number of ways, when all the books are arranged language wise) 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! − 7!4!5!. Hence, the answer is 16! − 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! + 7!4!5!.

Answer. 16! − 7! · 9 · 9! + 5! · 1111! + 4! · 12 · 12! + 7!4!5!.

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