Menu Close

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-

Tinku Tara 0 Comments
Pin




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

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *