All five letter words are made using all the letters A, B, C, D, … - Vidyadip

Question

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 $P \left( W _{ n }\right)$ of choosing the word $W _{ n }$ satisfy $P \left( W _{ n }\right)=2 P \left( W _{ n -1}\right), n >1$.
If $P ( CDBEA )=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in N$, then $\alpha+\beta$ is equal to : __________.

✓

Answer

(183)
Let $P\left(W_1\right)=x$
$
\begin{array}{l}
\sum_{i=1}^{120} P\left(W_{i}\right)=1 \\
x+2 x+2^2 x+2^3 x+\ldots+2^{119} x=1 \\
\frac{x\left(2^{120}-1\right)}{(2-1)}=1 \Rightarrow x=\frac{1}{2^{120}-1}\quad \quad \ldots \ldots(1)
\end{array}
$
Rank of CDBEA

$\begin{array}{l}\text { So, } P \left( W _{64}\right)=2 P \left( W _{63}\right)=\ldots=2^{63} P \left( W _1\right) \\ =\frac{2^{63}}{2^{120}-1} \\ \alpha+\beta=63+120=183\end{array}$

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