Let $n!_{\%p}$ be a special factorial where $n!$ is divided by the maximum exponent of $p$ that divides $n!$

$$n!_{\%p} = \frac{n!}{d_p(n!)}$$

Where $d_p(n!)$ is called Legendre’s Formula which is explained in detail in this article

Compute $n!_{\%p} \pmod{p}$ given that $p$ is a prime number

First let’s write this special factorial explicitely

$$\begin{equation} \label{explicit} n!_{\%p} = \tfrac{1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot p \cdot (p + 1) \cdot \ldots \cdot (2p - 1) \cdot 2p \cdot (2p + 1) \cdot \ldots \cdot (kp - 1) \cdot kp \cdot (kp + 1) \cdot \ldots \cdot (n - 1) \cdot n}{p^{ \tfrac{n}{p} + \tfrac{n}{p^2} + ... }} \end{equation}$$

The number $kp$ is a number that is divisible by $p$, we also see that $k$ might be a composite number that could be divisible by $p$ again

Now let’s first divide the equation by $p^{ \tfrac{n}{p} }$ which is exactly the number of multiples of $p$

$$n!_{\%p} = \tfrac{1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot 1 \cdot (p + 1) \cdot \ldots \cdot (2p - 1) \cdot 2 \cdot (2p + 1) \cdot \ldots \cdot (kp - 1) \cdot k \cdot (kp + 1) \cdot \ldots \cdot (n - 1) \cdot n}{p^{ \tfrac{n}{p^2} + ... }}$$

If we apply the modulo operation to each term except the multiples of $p$ we have

$$n!_{\%p} = \tfrac{1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot 1 \cdot 1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot 2 \cdot 1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot p \cdot 1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot kp}{p^{ \tfrac{n}{p^2} + ... }} \cdot 1 \cdot 2 \cdot \ldots \cdot (n - 1) \cdot n$$

NOTE: we’re not applying the modulo operator to each multiple of $p$ because they don’t actually exist since there are no $p$ factors in the equation, they are reduced with posterior divisions by $p^{ \tfrac{n}{p^i} }$

NOTE: the number $kp$ described in \eqref{explicit} just denotes a multiple of $p$

We see that the expression $1 \cdot 2 \cdot \ldots \cdot (p - 1)$ is repeated many times in the equation above + a product of some additional terms which don’t form an entire sequence, let $c = 1 \cdot 2 \cdot \ldots \cdot (p - 1)$ then

$$n!_{\%p} = \tfrac{1c \cdot 2c \cdot \ldots \cdot (p - 1)c \cdot pc \cdot (p + 1)c \cdot \ldots \cdot (kp - 1)c \cdot kpc}{p^{ \tfrac{n}{p^2} + ... }} \cdot 1 \cdot 2 \cdot \ldots \cdot (n - 1) \cdot n$$

Since each $c$ factor occurs in every contiguous sequence of length $p$ there are exactly $\left \lfloor \tfrac{n}{p} \right \rfloor$ $c$ factors, factoring $c$ we have

$$n!_{\%p} = c^{\left \lfloor \tfrac{n}{p} \right \rfloor} \cdot \tfrac{1 \cdot 2 \cdot \ldots \cdot (p - 1) \cdot p \cdot (p + 1) \cdot \ldots \cdot (2p - 1) \cdot 2p \cdot (2p + 1) \cdot \ldots \cdot (kp - 1) \cdot kp}{p^{ \tfrac{n}{p^2} + ... }} \cdot 1 \cdot 2 \cdot \ldots \cdot (n - 1) \cdot n$$

Note that the term multiplying $c^{\left \lfloor \tfrac{n}{p} \right \rfloor}$ is the same as \eqref{explicit}, we now have to divide it by $p^{ \tfrac{n}{p^2} }$ which is exactly the number of multiples of $p^2$ (NOTE: $kp$ is a multiple of $p$ but might/might not be a multiple of $p^2$)

This observation leads to a recursive implementation

Complexity: $O(p , log_p{n})$

long long special_factorial_mod_p(long long n, long long p) {
long long res = 1;

// computation of c
long long c = p-1;

while (n > 1) {
res = (res * binary_exponentiation_modulo_m(c, n / p, p)) % p;
for (long long i = 2; i <= n % p; i += 1) {
res = (res * (long long)i) % p;
}
n /= p;
}
return res % p;
}


## Applications

### Finding the value of $nCr % p$

We can quickly calculate the value of $nCr % p$, we can compute the maximum exponents of $p$ in $n!$, $(n - r)!$ and $r!$, let those numbers be $p^a$, $p^b$ and $p^c$ then $nCr$ can be expressed as

$$nCr = \binom{p^a \cdot \ldots}{p^b \cdot p^c \ldots}$$

Which means that $nCr$ will be a multiple of $p$ when $a - b - c > 0$, if $a - b - c = 0$ then the number is equal to

$$nCr = \frac{n!_{\%p}}{(n - r)!_{\%p} \cdot r!_{\%p}}$$

NOTE: $a - b - c$ can never be less than zero, that would imply that $nCr$ is not an integer

The denominator can be found using the modular multiplicative inverse of $(n - r)!_{\%p}$ and $r!_{\%p}$

long long nCr_mod_p(int n, int r, int p) {
int a = max_power_in_factorial(n, p);
int b = max_power_in_factorial(n - r, p);
int c = max_power_in_factorial(r, p);
if (a > b + c) {
return 0;
}

return (special_factorial_mod_p(n, p) *
((modular_multiplicative_inverse(special_factorial_mod_p(n - r, p), p) *
modular_multiplicative_inverse(special_factorial_mod_p(r, p), p)) % p) % p);
}