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Question Number 134498 by bemath last updated on 04/Mar/21

$$ \\ $$ Consider a 12-letter word made up of 8 b’s and 4 a’s. what is the probability that randomly shuffling its letters lets exactly two a’s come together and two other a’s be separated (as in the following example: baababbabbbb)?\\n

Answered by EDWIN88 last updated on 04/Mar/21

The required probability is equal to( the number of permutations with 2 a′s together and 2 a′s separated)divided by the total numbers of permutations (•) The permutations with 2 a′s together and 2 a′s separated can end with b or ba or baa . (1)The permutations which end with b can be formed by combining the character or string : aab, ab, ab and 5b′s . The number of these is C_1 ^( 8) × C_2 ^( 7) = 168 (2) The permutations which end in baa be formed by combining the characters or string : aab, ab 6b′s and the final a . The number of these is C_( 1) ^( 8) × C_1 ^( 7) = 56 (3) The permutations which end in baa can be formed by combining the characters of string ab, ab, 6b′s and the final aa. The number of these is C_2 ^( 8) = 28 (4)The total number of permutations is C_4 ^( 12) = 495 So the probability is ((168+56+28)/(495)) = ((28)/(55))

$$\mathrm{The}\:\mathrm{required}\:\mathrm{probability}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\left(\:\mathrm{the}\:\mathrm{number}\right. \\ $$ $$\mathrm{of}\:\mathrm{permutations}\:\mathrm{with}\:\mathrm{2}\:\mathrm{a}'\mathrm{s}\:\mathrm{together}\:\mathrm{and}\:\mathrm{2}\:\mathrm{a}'\mathrm{s} \\ $$ $$\left.\mathrm{separated}\right)\mathrm{divided}\:\mathrm{by}\:\mathrm{the}\:\mathrm{total}\:\mathrm{numbers}\:\mathrm{of} \\ $$ $$\mathrm{permutations} \\ $$ $$\color{mathblue}{\left(}\color{mathblue}{\bullet}\color{mathblue}{\right)}\color{mathblue}{\:}\mathrm{\color{mathblue}{T}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{p}\color{mathblue}{e}\color{mathblue}{r}\color{mathblue}{m}\color{mathblue}{u}\color{mathblue}{t}\color{mathblue}{a}\color{mathblue}{t}\color{mathblue}{i}\color{mathblue}{o}\color{mathblue}{n}\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{w}\color{mathblue}{i}\color{mathblue}{t}\color{mathblue}{h}}\color{mathblue}{\:}\mathrm{\color{mathblue}{2}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}}\color{mathblue}{'}\mathrm{\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{o}\color{mathblue}{g}\color{mathblue}{e}\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}\color{mathblue}{r}}\color{mathblue}{\:} \\ $$ $$\mathrm{\color{mathblue}{a}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{2}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}}\color{mathblue}{'}\mathrm{\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{s}\color{mathblue}{e}\color{mathblue}{p}\color{mathblue}{a}\color{mathblue}{r}\color{mathblue}{a}\color{mathblue}{t}\color{mathblue}{e}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{a}\color{mathblue}{n}}\color{mathblue}{\:}\mathrm{\color{mathblue}{e}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{w}\color{mathblue}{i}\color{mathblue}{t}\color{mathblue}{h}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}}\color{mathblue}{\:}\mathrm{\color{mathblue}{o}\color{mathblue}{r}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}\color{mathblue}{a}}\color{mathblue}{\:}\mathrm{\color{mathblue}{o}\color{mathblue}{r}} \\ $$ $$\mathrm{\color{mathblue}{b}\color{mathblue}{a}\color{mathblue}{a}}\color{mathblue}{\:}\color{mathblue}{.} \\ $$ $$\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\mathrm{\color{mathred}{T}\color{mathred}{h}\color{mathred}{e}}\color{mathred}{\:}\mathrm{\color{mathred}{p}\color{mathred}{e}\color{mathred}{r}\color{mathred}{m}\color{mathred}{u}\color{mathred}{t}\color{mathred}{a}\color{mathred}{t}\color{mathred}{i}\color{mathred}{o}\color{mathred}{n}\color{mathred}{s}}\color{mathred}{\:}\mathrm{\color{mathred}{w}\color{mathred}{h}\color{mathred}{i}\color{mathred}{c}\color{mathred}{h}}\color{mathred}{\:}\mathrm{\color{mathred}{e}\color{mathred}{n}\color{mathred}{d}}\color{mathred}{\:}\mathrm{\color{mathred}{w}\color{mathred}{i}\color{mathred}{t}\color{mathred}{h}}\color{mathred}{\:}\mathrm{\color{mathred}{b}}\color{mathred}{\:}\mathrm{\color{mathred}{c}\color{mathred}{a}\color{mathred}{n}}\color{mathred}{\:} \\ $$ $$\mathrm{\color{mathred}{b}\color{mathred}{e}}\color{mathred}{\:}\mathrm{\color{mathred}{f}\color{mathred}{o}\color{mathred}{r}\color{mathred}{m}\color{mathred}{e}\color{mathred}{d}}\color{mathred}{\:}\mathrm{\color{mathred}{b}\color{mathred}{y}}\color{mathred}{\:}\mathrm{\color{mathred}{c}\color{mathred}{o}\color{mathred}{m}\color{mathred}{b}\color{mathred}{i}\color{mathred}{n}\color{mathred}{i}\color{mathred}{n}\color{mathred}{g}}\color{mathred}{\:}\mathrm{\color{mathred}{t}\color{mathred}{h}\color{mathred}{e}}\color{mathred}{\:}\mathrm{\color{mathred}{c}\color{mathred}{h}\color{mathred}{a}\color{mathred}{r}\color{mathred}{a}\color{mathred}{c}\color{mathred}{t}\color{mathred}{e}\color{mathred}{r}}\color{mathred}{\:}\mathrm{\color{mathred}{o}\color{mathred}{r}} \\ $$ $$\mathrm{\color{mathred}{s}\color{mathred}{t}\color{mathred}{r}\color{mathred}{i}\color{mathred}{n}\color{mathred}{g}}\color{mathred}{\:}\color{mathred}{:}\color{mathred}{\:}\mathrm{\color{mathred}{a}\color{mathred}{a}\color{mathred}{b}}\color{mathred}{,}\color{mathred}{\:}\mathrm{\color{mathred}{a}\color{mathred}{b}}\color{mathred}{,}\color{mathred}{\:}\mathrm{\color{mathred}{a}\color{mathred}{b}}\color{mathred}{\:}\mathrm{and}\:\mathrm{\color{mathred}{5}\color{mathred}{b}}\color{mathred}{'}\mathrm{\color{mathred}{s}}\color{mathred}{\:}\color{mathred}{.}\:\mathrm{\color{mathred}{T}\color{mathred}{h}\color{mathred}{e}}\color{mathred}{\:}\mathrm{number} \\ $$ $$\mathrm{of}\:\mathrm{these}\:\mathrm{is}\:\mathrm{C}_{\mathrm{1}} ^{\:\mathrm{8}} \:×\:\mathrm{C}_{\mathrm{2}} ^{\:\mathrm{7}} \:=\:\mathrm{168} \\ $$ $$\color{mathblue}{\left(}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}\color{mathblue}{\:}\mathrm{\color{mathblue}{T}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{p}\color{mathblue}{e}\color{mathblue}{r}\color{mathblue}{m}\color{mathblue}{u}\color{mathblue}{t}\color{mathblue}{a}\color{mathblue}{t}\color{mathblue}{i}\color{mathblue}{o}\color{mathblue}{n}\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{w}\color{mathblue}{h}\color{mathblue}{i}\color{mathblue}{c}\color{mathblue}{h}}\color{mathblue}{\:}\mathrm{\color{mathblue}{e}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{i}\color{mathblue}{n}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}\color{mathblue}{a}\color{mathblue}{a}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}\color{mathblue}{e}} \\ $$ $$\mathrm{\color{mathblue}{f}\color{mathblue}{o}\color{mathblue}{r}\color{mathblue}{m}\color{mathblue}{e}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}\color{mathblue}{y}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{o}\color{mathblue}{m}\color{mathblue}{b}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{g}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{h}\color{mathblue}{a}\color{mathblue}{r}\color{mathblue}{a}\color{mathblue}{c}\color{mathblue}{t}\color{mathblue}{e}\color{mathblue}{r}\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{o}\color{mathblue}{r}} \\ $$ $$\mathrm{\color{mathblue}{s}\color{mathblue}{t}\color{mathblue}{r}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{g}}\color{mathblue}{\:}\color{mathblue}{:}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}\color{mathblue}{a}\color{mathblue}{b}}\color{mathblue}{,}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}\color{mathblue}{b}}\color{mathblue}{\:}\mathrm{\color{mathblue}{6}\color{mathblue}{b}}\color{mathblue}{'}\mathrm{\color{mathblue}{s}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{f}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{a}\color{mathblue}{l}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}}\color{mathblue}{\:}\color{mathblue}{.} \\ $$ $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{these}\:\mathrm{is}\:\mathrm{C}_{\:\mathrm{1}} ^{\:\mathrm{8}} \:×\:\mathrm{C}_{\mathrm{1}} ^{\:\mathrm{7}} \:=\:\mathrm{56} \\ $$ $$\:\left(\mathrm{3}\right)\:\mathrm{\color{mathbrown}{T}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{p}\color{mathbrown}{e}\color{mathbrown}{r}\color{mathbrown}{m}\color{mathbrown}{u}\color{mathbrown}{t}\color{mathbrown}{a}\color{mathbrown}{t}\color{mathbrown}{i}\color{mathbrown}{o}\color{mathbrown}{n}\color{mathbrown}{s}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{w}\color{mathbrown}{h}\color{mathbrown}{i}\color{mathbrown}{c}\color{mathbrown}{h}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{e}\color{mathbrown}{n}\color{mathbrown}{d}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{i}\color{mathbrown}{n}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{b}\color{mathbrown}{a}\color{mathbrown}{a}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{c}\color{mathbrown}{a}\color{mathbrown}{n}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{b}\color{mathbrown}{e}} \\ $$ $$\mathrm{\color{mathbrown}{f}\color{mathbrown}{o}\color{mathbrown}{r}\color{mathbrown}{m}\color{mathbrown}{e}\color{mathbrown}{d}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{b}\color{mathbrown}{y}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{c}\color{mathbrown}{o}\color{mathbrown}{m}\color{mathbrown}{b}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{g}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{c}\color{mathbrown}{h}\color{mathbrown}{a}\color{mathbrown}{r}\color{mathbrown}{a}\color{mathbrown}{c}\color{mathbrown}{t}\color{mathbrown}{e}\color{mathbrown}{r}\color{mathbrown}{s}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{o}\color{mathbrown}{f}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{s}\color{mathbrown}{t}\color{mathbrown}{r}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{g}} \\ $$ $$\mathrm{\color{mathbrown}{a}\color{mathbrown}{b}}\color{mathbrown}{,}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{a}\color{mathbrown}{b}}\color{mathbrown}{,}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{6}\color{mathbrown}{b}}\color{mathbrown}{'}\mathrm{\color{mathbrown}{s}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{a}\color{mathbrown}{n}\color{mathbrown}{d}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{f}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{a}\color{mathbrown}{l}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{a}\color{mathbrown}{a}}\color{mathbrown}{.}\color{mathbrown}{\:} \\ $$ $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{these}\:\mathrm{is}\:\mathrm{C}_{\mathrm{2}} ^{\:\mathrm{8}} \:=\:\mathrm{28} \\ $$ $$\left(\mathrm{4}\right)\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{permutations}\:\mathrm{is} \\ $$ $$\:\mathrm{C}_{\mathrm{4}} ^{\:\mathrm{12}} \:=\:\mathrm{495} \\ $$ $$ \\ $$ $$\mathrm{So}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{is}\:\frac{\mathrm{168}+\mathrm{56}+\mathrm{28}}{\mathrm{495}}\:=\:\frac{\mathrm{28}}{\mathrm{55}} \\ $$

Answered by mr W last updated on 04/Mar/21

□aa⊡a⊡a□ ⊡=one or more b′s □=zero or more b′s (1+x+x^2 +...)^2 (x+x^2 +x^3 +...)^2 =x^2 Σ_(k=0) ^∞ C_3 ^(k+3) x^k coef. of x^8 term is at k=6: C_3 ^(6+3) number of valid words: C_3 ^(6+3) ×((3!)/(2!))=252 total number of words: ((12!)/(4!8!))=495 p=((C_3 ^(6+3) ×((3!)/(2!)))/((12!)/(4!8!)))=((252)/(495))=((28)/(55))

$$\Box{aa}\boxdot{a}\boxdot{a}\Box \\ $$ $$\color{mathbrown}{\boxdot}\color{mathbrown}{=}{\color{mathbrown}{o}\color{mathbrown}{n}\color{mathbrown}{e}}\color{mathbrown}{\:}{\color{mathbrown}{o}\color{mathbrown}{r}}\color{mathbrown}{\:}{\color{mathbrown}{m}\color{mathbrown}{o}\color{mathbrown}{r}\color{mathbrown}{e}}\color{mathbrown}{\:}{\color{mathbrown}{b}}\color{mathbrown}{'}{\color{mathbrown}{s}} \\ $$ $$\color{mathblue}{\Box}\color{mathblue}{=}{\color{mathblue}{z}\color{mathblue}{e}\color{mathblue}{r}\color{mathblue}{o}}\color{mathblue}{\:}{\color{mathblue}{o}\color{mathblue}{r}}\color{mathblue}{\:}{\color{mathblue}{m}\color{mathblue}{o}\color{mathblue}{r}\color{mathblue}{e}}\color{mathblue}{\:}{\color{mathblue}{b}}\color{mathblue}{'}{\color{mathblue}{s}} \\ $$ $$\color{mathblue}{\left(}\mathrm{\color{mathblue}{1}}\color{mathblue}{+}{\color{mathblue}{x}}\color{mathblue}{+}{\color{mathblue}{x}}^{\mathrm{\color{mathblue}{2}}} \color{mathblue}{+}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{\right)}^{\mathrm{\color{mathblue}{2}}} \color{mathbrown}{\left(}{\color{mathbrown}{x}}\color{mathbrown}{+}{\color{mathbrown}{x}}^{\mathrm{\color{mathbrown}{2}}} \color{mathbrown}{+}{\color{mathbrown}{x}}^{\mathrm{\color{mathbrown}{3}}} \color{mathbrown}{+}\color{mathbrown}{.}\color{mathbrown}{.}\color{mathbrown}{.}\color{mathbrown}{\right)}^{\mathrm{\color{mathbrown}{2}}} ={x}^{\mathrm{2}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{3}} ^{{k}+\mathrm{3}} {x}^{{k}} \\ $$ $${coef}.\:{of}\:{x}^{\mathrm{8}} \:{term}\:{is}\:{at}\:{k}=\mathrm{6}:\:{C}_{\mathrm{3}} ^{\mathrm{6}+\mathrm{3}} \\ $$ $${number}\:{of}\:{valid}\:{words}:\:{C}_{\mathrm{3}} ^{\mathrm{6}+\mathrm{3}} ×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{252} \\ $$ $${total}\:{number}\:{of}\:{words}:\:\frac{\mathrm{12}!}{\mathrm{4}!\mathrm{8}!}=\mathrm{495} \\ $$ $${p}=\frac{{C}_{\mathrm{3}} ^{\mathrm{6}+\mathrm{3}} ×\frac{\mathrm{3}!}{\mathrm{2}!}}{\frac{\mathrm{12}!}{\mathrm{4}!\mathrm{8}!}}=\frac{\mathrm{252}}{\mathrm{495}}=\frac{\mathrm{\color{mathred}{2}\color{mathred}{8}}}{\mathrm{\color{mathred}{5}\color{mathred}{5}}} \\ $$

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