Q. In how many ways can 3 books on English and 2 books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together?
My solution:
Lay 3 English books in a row. There are 4 gaps in between and on the sides of English books. These 4 gaps can be filled by 2 Hindi books in 4C2 = 6 ways as follows;
1. E1 H1 E2 H2 E3
2. H1 E1 H2 E2 E3
3. H1 E1 E2 H2 E3
4. H1 E1 E2 E3 H2
5. E1 H1 E2 E3 H2
6. E1 E2 H1 E3 H2
But we can also arrange 3 English books in 3! ways and 2 Hindi books in 2! ways.
Thus total ways of arrangements should be = 4C2 × 3! × 2! = 6×6×2 = 72 ways.
I checked online and most answer are giving 4C2. Why arrangement of English books and Hindi books have been ignored?
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Answer:
Total number of arrangement =6!=720
Total number of arrangement while all the Hindi books are together =4!×3!=24×6=144
∴ The number of ways, in which books are arranged, while all the Hindi books are not together
=720−144=576
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