Can you improve the performance of this multinomial probability query?

I would like to have the probability that some N samples of equiprobable events in M ​​have no case that M has individual results that have more than one maximum value.

The code I am using in Mathematica 11 is:

Probability[Max[Array[x, bins]]<= max, Array[x, bins] [Distributed] 
             Multinomial distribution[balls, ConstantArray[1/bins, bins]]]

But, this becomes very slow with the cases that interest me. For example:

{balls, containers, max} = {100, 10, 20}

It has been running for several minutes on my Intel 4-core laptop reasonably fast.

Can performance be improved in any way to ensure that cases like this or a similar magnitude of values ​​end up in less time?