The sum and product of the mean and variance of a binomial distribution are $82.5$ and $1350$ respectively. They the number of trials in the binomial distribution is

Given, sum of mean and variance is $82.5$

Product of roots is $1350$

Now, Mean and variance are the roots of 

$x^2-\left(\text{sum of roots}\right)x+\text{product of roots}=0$

$\Rightarrow x^2-82.5x+1350=0$

$\Rightarrow x=60 \text{or} 22.5$

So, mean $=np=60$ and variance $=npq=22.5$

$\Rightarrow q=\frac{22.5}{60}=\frac{3}{8}$

So, $p=\frac{5}{8}$ and $n=\frac{60}{\frac{5}{8}}=96$