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Question Number 110498 by Aina Samuel Temidayo last updated on 29/Aug/20

How many ways can the letters in the  word MATHEMATICS be  rearranged such that the word formed  either starts or ends with a vowel, and  any three consecutive letters must  contain a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{word}\:\mathrm{MATHEMATICS}\:\mathrm{be} \\ $$$$\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word}\:\mathrm{formed} \\ $$$$\mathrm{either}\:\mathrm{starts}\:\mathrm{or}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\:\mathrm{vowel},\:\mathrm{and} \\ $$$$\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{letters}\:\mathrm{must} \\ $$$$\mathrm{contain}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

I have the options. 332640 is not part  of them.

$$\mathrm{I}\:\mathrm{have}\:\mathrm{the}\:\mathrm{options}.\:\mathrm{332640}\:\mathrm{is}\:\mathrm{not}\:\mathrm{part} \\ $$$$\mathrm{of}\:\mathrm{them}. \\ $$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

It′s not also part of the options.

$$\mathrm{It}'\mathrm{s}\:\mathrm{not}\:\mathrm{also}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{options}. \\ $$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

That′s part of the options. So can you  send the solution?

$$\mathrm{That}'\mathrm{s}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{options}.\:\mathrm{So}\:\mathrm{can}\:\mathrm{you} \\ $$$$\mathrm{send}\:\mathrm{the}\:\mathrm{solution}? \\ $$

Answered by mr W last updated on 29/Aug/20

we have four vowels: A,A,E,I  seven consonants: M,M,T,T,H,C,S    any three consecutive letters must  contain at least a vowel, that means  two vowels may separated by at most  two consonants.    a valid word should start with a vowel  or end with a vowel. let′s look the  first case: it starts with a vowel and  ends with a consonant. such a valid  word can be formed like:  ■□■□■□■♦  ■ stands for one of the 4 vowels  □ can be occupied by none, one or two       consonants  ♦ can be occupied by one or two       consonants  to occupy the three □ places and the  one ♦ place with seven consonants,  there are 4 ways, they are  ■X■XX■XX■XX  ■XX■X■XX■XX  ■XX■XX■X■XX  ■XX■XX■XX■X  or using generating function:  (1+x+x^2 )^3 (x+x^2 )  the coefficient of x^7  term is 4, that  means there are 4 ways to occupy  the places with the 7 consonants.    to arrange the 4 vowels there are  ((4!)/(2!)) ways. to arrange the seven   consonants there are ((7!)/(2!2!)) ways. so  we have 4×((4!)/(2!))×((7!)/(2!2!)) ways to form a  word which starts with a vowel and  ends with a consonant. we have  the same number of ways to form a  word which ends with a vowel and  starts with a consonant.    therefore totally we have  2×4×((4!)/(2!))×((7!)/(2!2!))=120960 ways to form  a word as requested.

$${we}\:{have}\:{four}\:{vowels}:\:{A},{A},{E},{I} \\ $$$${seven}\:{consonants}:\:{M},{M},{T},{T},{H},{C},{S} \\ $$$$ \\ $$$${any}\:{three}\:{consecutive}\:{letters}\:{must} \\ $$$${contain}\:{at}\:{least}\:{a}\:{vowel},\:{that}\:{means} \\ $$$${two}\:{vowels}\:{may}\:{separated}\:{by}\:{at}\:{most} \\ $$$${two}\:{consonants}. \\ $$$$ \\ $$$${a}\:{valid}\:{word}\:{should}\:{start}\:{with}\:{a}\:{vowel} \\ $$$${or}\:{end}\:{with}\:{a}\:{vowel}.\:{let}'{s}\:{look}\:{the} \\ $$$${first}\:{case}:\:{it}\:{starts}\:{with}\:{a}\:{vowel}\:{and} \\ $$$${ends}\:{with}\:{a}\:{consonant}.\:{such}\:{a}\:{valid} \\ $$$${word}\:{can}\:{be}\:{formed}\:{like}: \\ $$$$\blacksquare\Box\blacksquare\Box\blacksquare\Box\blacksquare\diamondsuit \\ $$$$\blacksquare\:{stands}\:{for}\:{one}\:{of}\:{the}\:\mathrm{4}\:{vowels} \\ $$$$\Box\:{can}\:{be}\:{occupied}\:{by}\:{none},\:{one}\:{or}\:{two} \\ $$$$\:\:\:\:\:{consonants} \\ $$$$\diamondsuit\:{can}\:{be}\:{occupied}\:{by}\:{one}\:{or}\:{two} \\ $$$$\:\:\:\:\:{consonants} \\ $$$${to}\:{occupy}\:{the}\:{three}\:\Box\:{places}\:{and}\:{the} \\ $$$${one}\:\diamondsuit\:{place}\:{with}\:{seven}\:{consonants}, \\ $$$${there}\:{are}\:\mathrm{4}\:{ways},\:{they}\:{are} \\ $$$$\blacksquare{X}\blacksquare{XX}\blacksquare{XX}\blacksquare{XX} \\ $$$$\blacksquare{XX}\blacksquare{X}\blacksquare{XX}\blacksquare{XX} \\ $$$$\blacksquare{XX}\blacksquare{XX}\blacksquare{X}\blacksquare{XX} \\ $$$$\blacksquare{XX}\blacksquare{XX}\blacksquare{XX}\blacksquare{X} \\ $$$${or}\:{using}\:{generating}\:{function}: \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} \left({x}+{x}^{\mathrm{2}} \right) \\ $$$${the}\:{coefficient}\:{of}\:{x}^{\mathrm{7}} \:{term}\:{is}\:\mathrm{4},\:{that} \\ $$$${means}\:{there}\:{are}\:\mathrm{4}\:{ways}\:{to}\:{occupy} \\ $$$${the}\:{places}\:{with}\:{the}\:\mathrm{7}\:{consonants}. \\ $$$$ \\ $$$${to}\:{arrange}\:{the}\:\mathrm{4}\:{vowels}\:{there}\:{are} \\ $$$$\frac{\mathrm{4}!}{\mathrm{2}!}\:{ways}.\:{to}\:{arrange}\:{the}\:{seven}\: \\ $$$${consonants}\:{there}\:{are}\:\frac{\mathrm{7}!}{\mathrm{2}!\mathrm{2}!}\:{ways}.\:{so} \\ $$$${we}\:{have}\:\mathrm{4}×\frac{\mathrm{4}!}{\mathrm{2}!}×\frac{\mathrm{7}!}{\mathrm{2}!\mathrm{2}!}\:{ways}\:{to}\:{form}\:{a} \\ $$$${word}\:{which}\:{starts}\:{with}\:{a}\:{vowel}\:{and} \\ $$$${ends}\:{with}\:{a}\:{consonant}.\:{we}\:{have} \\ $$$${the}\:{same}\:{number}\:{of}\:{ways}\:{to}\:{form}\:{a} \\ $$$${word}\:{which}\:{ends}\:{with}\:{a}\:{vowel}\:{and} \\ $$$${starts}\:{with}\:{a}\:{consonant}. \\ $$$$ \\ $$$${therefore}\:{totally}\:{we}\:{have} \\ $$$$\mathrm{2}×\mathrm{4}×\frac{\mathrm{4}!}{\mathrm{2}!}×\frac{\mathrm{7}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{120960}\:{ways}\:{to}\:{form} \\ $$$${a}\:{word}\:{as}\:{requested}. \\ $$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

Thanks.

$$\mathrm{Thanks}. \\ $$

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