Codeword Lengths.

Consider a lossless compression code that uses five binary codewords with lengths

$\{1,2,3,4,5\}.$

$($a$)$ Prove that a uniquely decodable code exists with these codeword lengths. While

you are at it$,$ prove that a prefix free code exists with these codeword lengths.

$($b$)$ Exhibit a prefix free code with these lengths.

$($c$)$ Either give a PMF for which your code is a Huffman code, or explain why your

code cannot be a Huffman code for any PMF$.$