Separate permutations of a string modulo simple

Question

I recently thought about the following problem, and I am very surprised that no one seemed to ask this question:

For a string, how many different permutations of this exist, modulo ?

I know the formula where is the length of the string, and is the count of each character (given the alphabet size <img src = "/img/8448f6036f8b3c1845222e9d41e42efe.gif" alt = "K"> ). Thus, the string toffeewould have various permutations.

But this no longer works when it can be really big (say ), because the calculation goes beyond the long long int, and using BigIntegers will be too slow. Is there a way to calculate this, for example, or time?

If I pre-processed the factorials from to , and my "strings" came in the form of an array of length , where each element contains a counter for each letter, could it be calculated at or time?

Would thank for any help on this :)

Answer 1

, p = 10^9 + 7 - . , :

n! / (a1!*...*ak!) = 

n! * a1!^(p - 2) * ... * ak!^(p - 2) (mod p)

mod p - ( ).

O(k log p + n), O(k) , - O(log p), n! .

, .

Answer 2

, , . , "" . , , , .

, , .

Answer 3

N ( , , , - Sieve of Eratosthenes), N! .

, , .

, ( ), , , .

Answer 4

.. . .

modulo 1000000007 ( ), . mod - O (N + K * log (mod)).