$($a$)$ Consider a word constructed of $9$ data bits. Each bit is either a $1$ or a $0.$ An example is $101101001.$

Further, let each data bit be zero with probability $0\mathrm{.}65,$ independent of any other data bit.

i$.$ How many different $($unique$)$ code words are there?

ii$.$ How many code words have exactly two zeros?

iii. How many code words have exactly three zeros?

iv$.$ What is the probability of code word $101101001?$

v$.$ What is the probability that the $9-$bit word contains exactly six ones?

$($b$)$ Rework Prob. $2$a with the following changes: suppose the $9-$bit word is the concatenation of an

even parity bit with $8-$bits of data. For even parity, the parity bit is set or cleared to ensure

that the total number of $1\text{'}$s in the $9-$bit result is an even number. For example, the $8-$bit data

$01101001$ has an even number of ones, so the parity bit is set to $0$ to ensure the $9-$bit result

$001101001$ has an even number of ones.