1. **Problem Statement:** Calculate the total number of distinct variable names recognized by the compiler based on the given rules. 2. **Rules for variable names:** - Numeric variable names: start with a letter, followed by either another letter, a digit, or nothing. - String variable names: start with the symbol $ followed by a letter, which can then be followed by another letter, a digit, or nothing. 3. **Define the sets:** - Let $L$ = number of letters. - Let $D$ = number of digits. Assuming the English alphabet, $L = 26$ and digits $D = 10$. 4. **Calculate numeric variable names:** - First character: must be a letter, so $26$ options. - Second character: can be a letter ($26$), a digit ($10$), or nothing ($1$ option). Total numeric variable names = $26 \times (26 + 10 + 1) = 26 \times 37 = 962$. 5. **Calculate string variable names:** - First character: fixed as $ (1 option). - Second character: must be a letter, $26$ options. - Third character: can be a letter ($26$), a digit ($10$), or nothing ($1$ option). Total string variable names = $1 \times 26 \times (26 + 10 + 1) = 26 \times 37 = 962$. 6. **Total distinct variable names:** Sum numeric and string variable names: $$962 + 962 = 1924$$ **Final answer:** The compiler recognizes **1924** distinct variable names.