I want to get values of factorial divisions such as 100!/(2!5!60!)(the numbers in the denominator will all be smaller than the numerator, and the sum of the denominators(numbers without factorial) can be at max 100) etc.
I read in a forum that a Pascal's triangle can be used for calculating such results, but a lot of searching didn't tell me how to do so.
It is actually for a programming problem, and I tried using the lgamma function to calculate the value, but that gives an error as I need a lot of precision.
So what could be used to calculate such factorial divisions?
Best Answer
You can express those in terms of binomial coefficients and factorials.
$$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$
For example
$$\frac{100!}{60! 5! 2!} = \frac{100!}{60! 40!} \cdot \frac{40!}{5! 35!} \cdot \frac{35!}{2! 33!} \cdot 33! = \binom{100}{60} \cdot \binom{40}{5} \cdot \binom{35}{2} \cdot 33!$$
Or more generally, if $a_1 \ge a_2 \ge \cdots \ge a_k$ and $a_1 + \cdots + a_k \le n$, we have
$$\frac{n!}{a_1! a_2! \ldots a_k!} = \binom{n}{a_1} \cdot \binom{n-a_1}{a_2} \ldots \binom{n-a_1-\cdots-a_{n-1}}{a_n} \cdot (n-a_1-\cdots-a_{n})!$$
And binomial coefficients are easily computed (without division) using Pascal's triangle relation.