Permutations And Combinations

Question
CBSEENMA11014105

How many words with or without meaning can be formed using all the letters of the word ‘EQUATION’ using each letter exactly once?

Additional part: How many of these words begin with E and end with N?

Solution

The word is 'EQUATION'
         Number of letters = 8 (all distinct)
     Number of letters to be used = 8
rightwards double arrow          Number of permutations = straight P presuperscript 8 subscript 8 = 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
Additional part:
     Fix E in box 1 and N in box 8
               
Number of permutations for box 1 = space space straight P presuperscript 1 subscript 1 space equals space 1                                                         ...(i)
Number of permutations for box 8 = straight P presuperscript 1 subscript 1 equals 1                                                             ...(ii)
  
                       Number of letters left = 6 rightwards double arrow n = 6
                       Number of boxes left = 6   rightwards double arrow r = 6
∴         Number of permutations = straight P presuperscript straight n subscript straight r space equals space straight P presuperscript 6 subscript 6 space equals space fraction numerator 6 factorial over denominator 0 factorial end fraction equals space 6 factorial space equals space 6 space straight x space 5 space straight x space 4 space straight x space 3 space straight x space 2 space straight x space 1 space equals space 720 ...(iii)
From (i), (ii) and (iii), by fundamental principle of counting, the toal number of words formed
                                          = 1 x 1 x 720 = 720.

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