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 proba

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 $P (W_{n})$ of choosing the word $W_{n}$ satisfy $P (W_{n}) = 2 P (W_{n - 1}), n > 1$ .
If $P (CDBEA) = \frac{2 ^{α}}{2 ^{β} - 1}, α, β \in N$ , then $α + β$ is equal to : _______

TestHub · Accepted Answer

Let P ( W 1 ​ ) = x ​ i = 1 ∑ 120 ​ P ( W i ​ ) = 1 x + 2 x + 2 2 x + 2 3 x + … + 2 119 x = 1 ( 2 − 1 ) x ( 2 120 − 1 ) ​ = 1 ⇒ x = 2 120 − 1 1 ​ ... ( i ) ​ Rank of CDBEA ​ C D B A E = 1 C D B E A = 1 ​ ​ So, P ( W 64 ​ ) = 2 P ( W 63 ​ ) = … = 2 63 P ( W 1 ​ ) = 2 120 − 1 2 63 ​ α + β = 63 + 120 = 183 ​

Mathematics - Probability Question with Solution | TestHub

Doubts & Discussion