Permutations And Combinations


How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that no two consonants are together.


Number of letters = (all distinct)
Number of vowels = 5 (e, i, o, u, a)
Number of consonants = 3 (q, t, n)
Arrange the vowels in a row by leaving one place between every two vowels.
Number of permutations of arrranging vowels <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/ at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/ line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>                   ...(i)
circled times    V    circled times    V     circled times   V   circled times    V    circled times   V     circled times
 1             2                 3           4            5            6
Number of places for the consonants, so that no two of them are together = 6
Number of consonants = 3
rightwards double arrow                           n = 6,  r = 3
Number of permutations of arranging consonants:
                           equals straight P presuperscript 6 subscript 3 space equals space fraction numerator 6 factorial over denominator 3 factorial end fraction space equals space 6 space straight x space 5 space straight x space 4 equals 120                           ...(ii)
Hence, using (i) and (ii), the number of words formed so that no two consonants are together, using fundamental principle of counting
                             = 120 x 120 = 14400

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