Calculation of binomial coefficients ... manually

Earlier in two articles, the topic of computing binomial coefficients using a computer was raised.
Calculation of Binomial Coefficients in C (C ++)
Calculation of Binomial Coefficients Using Fourier Transforms
After reading them, one may think that this is a complex and resource-intensive task.
Before programming something, let’s try to figure out what's what.

Factorial formula:

Let us open it:

Obviously, then

And now, let's try to calculate for example :



It is immediately clear that 10, 9 and 8 are mutually reduced, leading us to a

closer look, we see that the reductions do not end there. 14 is divided by 7, 12 by 6, 15 by 5, 16 by 4. The remaining 15 from three is divided by three, and the remaining from 7 2 is divided by the last two. Having made all these reductions, we get rid of the denominator, we get the following product:

Let's try to calculate the



First reduction:

Next, it is easy to see that 114/57 = 2, 112/56 = 2, 110/55 = 2, and so on, to 62/31 = 2

The second abbreviation:

So, we got 27 deuces in the numerator, which we will try to reduce with the denominator. First, cross out all the powers of two from the denominator and the corresponding number of twos from the numerator. (2, 4, 8, 16 = 10 deuces, 17 left)



The denominator has 11 more even numbers, some of them are evenly multiple. After all reductions, the denominator does not leave even numbers, and the numerator will have two deuces.



Let us now take the triples.


Well and further, we reduce by 5, 7, 11, 17, 19, 23, 29

It remains:

which is 23 544 809 717 399 465 172 377 319 715 292 496 .