For a language $L,$ let perm $(L)$ be the language:

$\{w$:xinL,$|w|=|x|,$ and $wis$ a permutation $of$ the letters $inx\}.$

$($a$)$ If $L$ is a Turing$-$decidable language, must perm $(L)$ be Turing$-$decidable? Explain.

$($b$)$ If $L$ is a Turing$-$recognizable language, must perm $(L)$ be Turing$-$recognizable? Explain.

Hint: Use an enumerator for $L.$

$($c$)$ Is there a language $L$ for which perm $(L)$ is not Turing$-$recognizable? Explain. Hint.

Question $51$ might help.

$($d$)$ If perm $(L)$ is a regular language, must $L$ be Turing$-$recognizable? Explain.