A text is made up of the characters a, b, c, d, e each occurring with the probability 0.11, 0.40, 0.16, 0.09 and 0.24 respectively. The optimal Huffman coding technique will have the average length of:Group of answer choices2.162.402.262.15

To solve this problem, we need to use the Huffman coding algorithm. This algorithm is used for lossless data compression and it's based on the principle of assigning shorter codes to more frequently used characters.

Here are the steps to solve this problem:

1. First, list all the characters and their probabilities:

   a: 0.11 b: 0.40 c: 0.16 d: 0.09 e: 0.24

2. Then, sort these characters based on their probabilities in ascending order:

   d: 0.09 a: 0.11 c: 0.16 e: 0.24 b: 0.40

3. Now, start building the Huffman tree from the bottom up. Combine the two characters with the smallest probabilities and create a new node with a probability that's the sum of the two. Repeat this process until you have a single node left:

   da: 0.20 c: 0.16 e: 0.24 b: 0.40

   cda: 0.36 e: 0.24 b: 0.40

   ecda: 0.60 b: 0.40

   becda: 1.00

4. Assign binary codes to each edge of the tree, starting with 0 for the left edge and 1 for the right edge. Then, to get the Huffman code for a character, you simply concatenate the binary codes of the edges on the path from the root to the character.

5. Finally, to get the average length of the Huffman codes, you multiply the probability of each character by the length of its Huffman code and sum up these values.

Let's assume the lengths of Huffman codes for the characters are as follows:

d: 4 a: 4 c: 3 e: 2 b: 1

Then, the average length would be:

(0.09 * 4) + (0.11 * 4) + (0.16 * 3) + (0.24 * 2) + (0.40 * 1) = 0.36 +