P$15$: A source produces cLaracters in $S=\{a,b,c,d\}$ in an IID fastion witl probabilities:

$uar{r}_{a}uar{r}_{b}uar{r}_{c}uar{r}_d>0$

If $T_a=uar{r}_c+uar{r}_d$ pove that a Huffman code exists witt all codiwnods of equal lengtls. Pive also thot another Huffmon code exists witl length $1,2,3$&$3.$

Find $uar{r}_{\text{m}\text{a}\text{x}}$ : The langest value of $f_a$ fre whick $T_a=uar{r}_c+uar{r}_d$ is possible. $uar{r}_a=uar{r}_c+uar{r}_d$ is possible.

Stow that if $f_a>{?}_{\text{m}\text{a}\text{x}}$ every optimal prefix$-$free code las a codiund of lengtl $1.$

Show if $n{f}_a$ all codawords Lave lingtts $2in$ every Huffmen code

$6$