(a) To find \(P(X = 3)\), we calculate the probability of opening two non-carrot tins followed by a carrot tin. The probability of the first tin not being carrots is \(\frac{4}{7}\) (since there are 4 non-carrot tins out of 7 total). The probability of the second tin not being carrots is \(\frac{3}{6}\) (since there are now 3 non-carrot tins out of 6 remaining). The probability of the third tin being carrots is \(\frac{3}{5}\) (since there are 3 carrot tins out of 5 remaining). Therefore, \(P(X = 3) = \frac{4}{7} \times \frac{3}{6} \times \frac{3}{5} = \frac{6}{35}\).

(b) The probability distribution table for \(X\) is:

(c) To find \(\text{Var}(X)\), we first calculate \(E(X)\):

\(E(X) = 1 \times \frac{15}{35} + 2 \times \frac{10}{35} + 3 \times \frac{6}{35} + 4 \times \frac{3}{35} + 5 \times \frac{1}{35} = \frac{70}{35} = 2\)

Then, \(\text{Var}(X)\) is calculated as:

\(\text{Var}(X) = \frac{1^2 \times 15 + 2^2 \times 10 + 3^2 \times 6 + 4^2 \times 3 + 5^2 \times 1}{35} - 2^2\)

\(= \frac{15 + 40 + 54 + 48 + 25}{35} - 4 = \frac{182}{35} - 4 = \frac{6}{5}\)