Find the average length of an optimum decipherable binary code for a source that emits five words with probabilities

$$0.25,0.15,0.15,0.2,0.25$$

Show that, if a source emits $N$ words (with $N \geqslant 2$ ), and if $l_{1}, \ldots, l_{N}$ are the lengths of the codewords in an optimum encoding over the binary alphabet, then

$$l_{1}+\cdots+l_{N} \leqslant \frac{1}{2}\left(N^{2}+N-2\right) .$$

[You may assume that an optimum encoding can be given by a Huffman encoding.]