Permutations And Combinations


How many words can be formed by using the letters of the word ‘ORIENTAL’ so that A and E always occupy the odd places?


Number of letters in word 'ORIENTAL' = 8 (all distinct)
Number of letters to be used = 8

Step I:      A and E are to occupy odd places marked X
                       Number of letters = 2 (A and E)
                        Number of boxes = 4
     rightwards double arrow                             n = 4, r = 2
      Number of permutations of arranging A and E = straight P presuperscript straight n subscript straight r space equals space straight P presuperscript 4 subscript 2 space equals space fraction numerator 4 factorial over denominator 2 factorial end fraction equals space 4 space cross times space 3 space equals space 12  ...(i)
Step II:  After A and E are fixed, there will be 6 letters left and 6 boxes for them.
             (A and E use up two boxes)
             rightwards double arrow Number of permutations of arrranging the remaining letters =space space space space space space straight P presuperscript 6 subscript 6 equals space space 6 factorial space equals space 720  ...(ii)
              Hence, from (i) and (ii), the total number of words formed = 12 x 720 = 8640.

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