All five letter words are made using all the letters A , B , C , D , E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by W_n . Let the probability P ( W _ n ) of choosing the word W

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All five letter words are made using all the letters A , B , C , D , E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by W_n . Let the probability P ( W _ n ) of choosing the word W _ n satisfy P ( W _ n )=2 P ( W _ n -1 ), n 1 . If P ( CDBEA )= 2^ 2^ -1 , , N , then + is equal to : _______

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Let P ( W ₁ )= x aligned & _ i=1 ¹²⁰ P (W_i )=1 & x+2 x+2^2 x+2^3 x+ +2¹¹⁹ x=1 & x (2¹²⁰-1 ) (2-1) =1 x= 1 2¹²⁰-1 ...(i) aligned Rank of CDBEA aligned & C D B A E =1 & C D B E A =1 aligned aligned & So, P ( W ₆₄ )=2 P ( W ₆₃ )= =2⁶³ P ( W ₁ ) & = 2⁶³ 2¹²⁰-1 & + =63+120=183 aligned

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