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91. Find the number of ways in which n books on mathematics and n books on chemistry can be placed alternatively on a shelf.

Total number of books = n + n = 2n

Let the places on the shelf for these books are: 1, 2, 3, 4, 5, 6, ..........(2n - 3), (2n - 2), (2n - 1), 2n.


M C M C M C M C M C

Case (i)

Mathematics books are arranged on odd number places and chemistry books are placed on even number places.

Number of arrangements = $straight P presuperscript straight n subscript straight n space cross times space straight P presuperscript straight n subscript straight n space equals space left parenthesis straight n factorial right parenthesis space left parenthesis straight n factorial right parenthesis$

Or

Case (ii)

Chemistry books are placed an odd number places and mathematics books on even number places.

Number of arrangements = $straight P presuperscript straight n subscript straight n space cross times space straight P presuperscript straight n subscript straight n space equals space left parenthesis straight n factorial right parenthesis space left parenthesis straight n factorial right parenthesis$

Hence, the total number of arrangements on the shelf where mathematics books and chemistry books are placed alternatively.

   $equals space left parenthesis straight n factorial right parenthesis space left parenthesis straight n factorial right parenthesis space plus space left parenthesis straight n factorial right parenthesis space left parenthesis straight n factorial right parenthesis space equals space left parenthesis straight n factorial right parenthesis squared space space plus space left parenthesis straight n factorial right parenthesis squared space equals space 2 space left parenthesis straight n factorial right parenthesis squared$