jmc

We could do this with a bit of casework, but that gets boring after a while. Instead, we can use complementary probability -- we'll find the probability that Mr. Wong has exactly the same number of grandsons as granddaughters, and then subtract this from 1. Since each grandchild can be male or female with equal likelihood, there are $2^{10}=1024$ possible ways in which the genders of the grandchildren can be determined. The only way in which Mr. Wong won't have more grandsons than granddaughters or more granddaughters than grandsons is if he has exactly 5 of each, which can occur in $\left(\genfrac{}{}{0ex}{}{10}{5}\right)$ ways, since there are $\left(\genfrac{}{}{0ex}{}{10}{5}\right)$ ways to choose 5 of the 10 children to be boys (the others then are girls). Therefore, the probability that Mr. Wong has the same number of grandsons and granddaughters is $$\frac{\left(\genfrac{}{}{0ex}{}{10}{5}\right)}{2^{10}}=\frac{252}{1024}=\frac{63}{256}.$$

Since the probability that he has the same number of granddaughters and grandsons is $\frac{63}{256}$, the probability he doesn't have the same number of granddaughters and grandsons is $1-\frac{63}{256}=\boxed{\frac{193}{256}}$.