Suppose that we are sending length $5$ binary words w $=$ b$1$b$2$b$3$b$4$b$5$ with bi in $\{0,1\}=$ F$2$ through a noisy binary symmetric channel $($BSC$)$ having error probability p for each bit sent.$($a$)(5$ points$)$ Compute the probability of at least one error during transmission, as a function of p$.$Now we choose to send w with two extra parity check bits b$6,$b$7$ as follows:wheref$($w$)=$ b$1$b$2$b$3$b$4$b$5$b$6$b$7$ b$6$ :$=$ b$1+$ b$2,$b$7$ :$=$ b$3+$ b$4+$ b$5.($b$)(10$ points$)$ Compute the probability of at least one undetectederror when w is sent as f$($w$),$ again as a function of p$.($c$)(5$ points$)$ Considering the image of f as a set of codewords C inside $\{0,1\}$ of length $7,$ what is the $($binary$)$ rate of the code C$?$