Explanation:

The given codes are $\text{AS}=19$ and $\text{BAT}=40$. We need to find the code for $\text{DEAF}$. Let's determine the rule by analyzing the given examples using positional values of letters (A=1, B=2, ... Z=26): For $\text{AS}$: $\text{A}=1, \text{S}=19$. If we sum them: $1+19 = 20 \neq 19$. If we multiply them: $1 \times 19 = 19$. This matches the given code. For $\text{BAT}$: $\text{B}=2, \text{A}=1, \text{T}=20$. If we sum them: $2+1+20 = 23 \neq 40$. If we multiply them: $2 \times 1 \times 20 = 40$. This matches the given code. The rule is to multiply the positional values of all letters in the word. Now, apply this rule to $\text{DEAF}$: $\text{D}=4, \text{E}=5, \text{A}=1, \text{F}=6$. Product $= 4 \times 5 \times 1 \times 6 = 20 \times 6 = 120$. Let's analyze the options: A) $180$: Incorrect, as $4 \times 5 \times 1 \times 6 = 120$. B) $190$: Incorrect. C) $160$: Incorrect. D) $120$: This option perfectly matches the derived rule of multiplying the positional values of the letters.