One card is drawn each from a $k$ deck of 52 cards where $k$ is a multiple of $52$.
I need to prove that information of an outcome where each card appears the same number of times tends to $\frac{51}{2} \log(k) + O(1)$ as $k\rightarrow \infty$ using Stirling's approximation.
Assuming $k=52m,$ I computed the probability of the required outcome as $$ p=\frac{{52m \choose m} {51m \choose m} \ldots {m \choose m}}{{52m \choose 52}} $$
So the information associated is $\log(p)$ but I am unable to derive the required result. Can anyone help me with this ?