Total number of rational terms in expansion of \((\frac{51}{12} + \frac{71}{18})^{360}\) is 11.

To find the total number of rational terms in the expansion, we need to consider the binomial expansion of the given expression and count the terms that are rational.

1. Binomial Expansion:
The given expression \((\frac{51}{12} + \frac{71}{18})^{360}\) can be expanded using the binomial theorem. According to the binomial theorem, the expansion of \((a + b)^n\) is given by:
\((a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \ldots + \binom{n}{n}a^0 b^n\)

In our case, \(a = \frac{51}{12}\), \(b = \frac{71}{18}\), and \(n = 360\).

2. Rational Terms:
To determine which terms in the expansion are rational, we need to consider the binomial coefficients \(\binom{n}{r}\) in each term. The binomial coefficient \(\binom{n}{r}\) is rational if and only if \(r\) is between 0 and \(n\) (inclusive).

3. Calculating Binomial Coefficients:
We need to calculate the binomial coefficients for \(r = 0\) to \(r = 360\) and check if each coefficient is rational or not.

4. Rational Binomial Coefficients:
The binomial coefficients that are rational are:
\(\binom{360}{0}, \binom{360}{1}, \binom{360}{2}, \binom{360}{3}, \binom{360}{4}, \binom{360}{5}, \binom{360}{6}, \binom{360}{7}, \binom{360}{8}, \binom{360}{9}, \binom{360}{10}\)

5. Counting Rational Terms:
There are 11 rational terms in the expansion.

6. Explanation:
The rational terms in the expansion occur when the powers of \(a\) and \(b\) combine to give a rational number. In this case, the powers in the binomial coefficients range from 0 to 10, resulting in 11 rational terms.

Therefore, the correct answer is 11.