Probability
Lexicographic word probability
nta_pyq_2025_apr
Grade 12
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 . Let the probability P (W ) of choosing the n n word W satisfy P (W$) = 2P$(W n n$n-1$) ,$n > 1$.$\alpha$If$P(CDBEA) = 2$$\beta$,$\alpha$,$\beta$$\ in $N , then$\alpha$+$\beta$is equal to : _______$2 -1$
Step-by-Step Solution
Key Concept: Compute the favourable cases for lexicographic word probability and divide by the total equally likely cases.
Let P (W$) = x$1 (183) 120$\sum$P (Wi$) = 1$$i=1$2 3 119$x + 2x + 2$$x + 2$x +$\ldots$+ 2$x = 1$120 x$(2 - 1)$$1 = 1$$\Rightarrow$$x = .$. . (i) 120$(2 - 1)$$2 - 1$Rank of CDBEA C D B A$E = 1$C D B E$A = 1$63 So, P ($W_{64}$$) = 2P$($W_{63}$) =$\ldots$= 2 P ($W_{1}$) 63$2 = 120$$2 - 1$$\alpha$+$\beta$=$63 + 120 = 183$
Correct Answer: 183