A message is made up entirely of characters from the set $X = \{P, Q, R, S, T\}$. The table of probabilities for each of the characters is shown below: If a message of 100 characters over $X$ is encoded using Huffman coding, then the expected length of the encoded message in bits is ________.

To find the expected length using Huffman coding, we construct the Huffman tree for the given probabilities:

Sorted initial frequencies:

• $T: 0.08$

• $R: 0.17$

• $S: 0.19$

• $P: 0.22$

• $Q: 0.34$

Step 1: Combine the two lowest probabilities $T (0.08)$ and $R (0.17)$ into a new node $N_1$ with probability $0.08 + 0.17 = 0.25$.

• Remaining: $S(0.19), P(0.22), N_1(0.25), Q(0.34)$

Step 2: Combine the next two lowest probabilities $S (0.19)$ and $P (0.22)$ into node $N_2$ with probability $0.19 + 0.22 = 0.41$.

• Remaining: $N_1(0.25), Q(0.34), N_2(0.41)$

Step 3: Combine $N_1 (0.25)$ and $Q (0.34)$ into node $N_3$ with probability $0.25 + 0.34 = 0.59$.

• Remaining: $N_2(0.41), N_3(0.59)$

Step 4: Combine $N_2 (0.41)$ and $N_3 (0.59)$ to form the root with probability $1.00$.

Codeword lengths:

• Length of $S$: 2 bits (depth in tree)

• Length of $P$: 2 bits

• Length of $Q$: 2 bits

• Length of $T$: 3 bits

• Length of $R$: 3 bits

Expected length per character ($L$):

For a message of 100 characters, the expected total length is:

Hence, the correct answer is 225.