Is it possible to calculate $\sum_{i=1}^n \ln(x_i!)$, given $\sum_{i=1}^n x_i$

Question

I have a problem solving

$$\sum_{i=1}^n \ln(x_i!)$$

The question gives me only

1. $\sum_{i=1}^n x_i$= $A$

2. Number of samples =$n$

I tried to transform the term but failed. Please help me solve this problem

Thank you for your help.

Answer 1

No, we can't find the sum of the logarithms of the factorials from the given information. For instance, if $A = n$, then all the $x_i$ could be $1$, which would make $$\sum_{i = 1}^n\ln(x_i!) = 0$$ or we could have $x_1 = n$ and all the other $x_i = 0$, which would give $$\sum_{i = 1}^n \ln(x_i!) = \ln(n!)$$