Question

How to simply write what I just wrote, I don’t even know what to search [discrete math]

Answer 1

Well that is hugely big.

You can alternatively express it as 19C10 × 18C9 × 17C8 × ... × 10C1

So you can break that down as

19! / (10!9!) × 18! / (9!9!) × ... × 10! / (9!1!)

There could be some simplification but it doesn't seem like it will be anything concise.

Answer 2

there are a few ways, depending what you plan to do with it.

```
Π(n=1→10)( n choose 9 )

Π(n=1→10)( (20-n)! / 9!10! )

aCb := a! / b!(a-b)!
→ (19c9)(18c9)...(10c9)
= 19!/9!10! × 18!/9!10! × ... × 10!/9!10!
= 19!18!17!...10! / 9!¹⁰10!¹⁰
= 19!18!...11! / 9!¹⁰10!⁹
```

there are definitely others but these seem most useful?

Answer 3

How about

                Π[a=10..19] nCr(a, 9)
              = Π[a=10..19] a! / (a-9)! / 9!
    = 1/9!^10 · Π[a=10..19] Π[b=0..9] (a-b)

Basically, find all pairwise differences between natural [10,19] and [0,9] and multiply all of them together