Understanding the Coding Problem

This problem requires us to find the correct code for a given sequence of numbers and symbols, '@ 9 7 8 3'. We need to use a provided table of letter codes and apply specific logical conditions that might alter the direct coding.

Coding Table Reference

Sequence Analysis and Base Code Generation

The sequence we need to code is: '@ 9 7 8 3'.

First, let's determine the direct letter code for each element in the sequence using the provided table:

• The symbol '@' corresponds to the code 'W'.
• The number '9' corresponds to the code 'C'.
• The number '7' corresponds to the code 'Z'.
• The number '8' corresponds to the code 'T'.
• The number '3' corresponds to the code 'L'.

Combining these, the initial or base code for the sequence '@ 9 7 8 3' is 'WCZTL'.

Applying Coding Conditions

Next, we must examine the given conditions to see if they apply to our sequence and modify the base code 'WCZTL'.

Evaluating Condition (i)

Condition (i): States that if the first element is a symbol and the last element is a number, their codes should be interchanged.

Let's check our sequence '@ 9 7 8 3':

• The first element is '@', which is indeed a symbol.
• The last element is '3', which is indeed a number.

Since both parts of this condition are met, we need to interchange the codes of the first ('W') and the last ('L') elements in our base code 'WCZTL'.

Interchanging 'W' and 'L' results in the new code: 'LCZTW'.

Evaluating Condition (ii)

Condition (ii): States that if the first element is an odd number and the last element is an even number, the codes for both the first and last elements should be replaced with '©'.

In our sequence '@ 9 7 8 3':

• The first element is '@', which is a symbol, not an odd number.

Because the first element is not an odd number, this condition does not apply to our sequence.

Evaluating Condition (iii)

Condition (iii): States that if both the second and third elements are perfect squares, the code for the third element should be changed to match the code for the second element.

Let's examine the second and third elements of '@ 9 7 8 3':

• The second element is '9'. We check if it's a perfect square: $9 = 3^2$. Yes, '9' is a perfect square.
• The third element is '7'. We check if it's a perfect square. '7' is not a perfect square.

Since the condition requires *both* the second and third elements to be perfect squares, and '7' is not, this condition does not apply.

Final Code Determination

After analyzing all the conditions, we found that only Condition (i) applied to the sequence '@ 9 7 8 3'. Applying this condition involved interchanging the codes of the first ('W') and last ('L') elements of the base code 'WCZTL'.

Therefore, the final correct code for the sequence is 'LCZTW'.