Permutations And Combinations

Question
CBSEENMA11014123

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

Solution

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

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