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How-many-ways-can-the-letters-in-the-word-MATHEMATICS-be-rearranged-such-that-the-word-formed-neither-starts-nor-ends-with-a-vowel-and-any-four-consecutive-letters-must-contain-at-least-a-vowel-




Question Number 110729 by mr W last updated on 30/Aug/20
How many ways can the letters in  the word MATHEMATICS  be rearranged such that the word  formed neither starts nor ends with a   vowel, and any four consecutive   letters must contain at least a vowel?
HowmanywayscanthelettersinthewordMATHEMATICSberearrangedsuchthatthewordformedneitherstartsnorendswithavowel,andanyfourconsecutivelettersmustcontainatleastavowel?
Answered by mr W last updated on 30/Aug/20
(see also Q110498)    n_1 ■n_2 ■n_3 ■n_4 ■n_5   ■ = vowel  n_(1...5) =number of consonants  1≤n_1 ,n_5 ≤3  0≤n_2 ,n_3 ,n_4 ≤3  n_1 +n_2 +n_3 +n_4 +n_5 =7  in generating function  (x+x^2 +x^3 )^2 (1+x+x^2 +x^3 )^3   =x^2 (1+x+x^2 )^2 (1+x+x^2 +x^3 )^3   =((x^2 (1−x^3 )^2 (1−x^4 )^3 )/((1−x)^5 ))  =x^2 (1−x^3 )^2 (1−x^4 )^3 Σ_(k=0) ^∞ C_4 ^(k+4) x^k   the coefficient of x^7  term is:  C_4 ^(5+4) −2×C_4 ^(2+4) −3×C_4 ^(1+4) =81    ⇒81×((7!)/(2!2!))×((4!)/(2!))=1 224 720 ways
(seealsoQ110498)n1◼n2◼n3◼n4◼n5◼=voweln15=numberofconsonants1n1,n530n2,n3,n43n1+n2+n3+n4+n5=7ingeneratingfunction(x+x2+x3)2(1+x+x2+x3)3=x2(1+x+x2)2(1+x+x2+x3)3=x2(1x3)2(1x4)3(1x)5=x2(1x3)2(1x4)3k=0C4k+4xkthecoefficientofx7termis:C45+42×C42+43×C41+4=8181×7!2!2!×4!2!=1224720ways

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