A language has a total of $8$ alphabets $\backslash$alpha $0,\backslash$alpha $1,\backslash$alpha $2,\backslash$alpha $3,\backslash$alpha $4,\backslash$alpha $5,\backslash$alpha $6,$ and $\backslash$alpha $7$ where these alphabets

occur in the language with probability p$,$ p$,2$p$,4$p$,8$p$,16$p$,32$p$,$ and $64$p respectively. A system

is being developed to represent these alphabets in a digital representation. Different digital

representations A$,$ B$,$ C$,$ and D are being considered as listed below.

For each digital representation listed below, please find the average number of bits required to

represent the alphabets in the language.

a$)$ Representation A: $3$ bits are allocated for each alphabet with the digital representations for the

$8$ alphabets given by $000,001,010,011,100,101,110,$ and $111$

b$)$Representation B: $2$ bits are allocated for alphabets $\backslash$alpha $0,\backslash$alpha $1,\backslash$alpha $2,$ and $\backslash$alpha $3$ and $3$ bits are allocated

for the alphabets $\backslash$alpha $4,\backslash$alpha $5,\backslash$alpha $6,$ and $\backslash$alpha $7$