# [isabelle] Binomial Coefficient as Factorial

```Dear all,

Hi, in Isabelle the definition we have for Binomial Coefficient is:

primrec binomial :: "nat â nat â nat" (infixl "choose" 65)

where

"0 choose k = (if k = 0 then 1 else 0)"

| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose
k))"

But according to my (limited) number theory knowledge, n choose k can also
be calculated using factorials and n choose k = n! / k! *(n-k)!.

So I want to prove the following lemma:

fun fac :: "nat â nat" where

"fac 0 = Suc 0"

| "fac n = n * fac (n - 1)"

lemma bin_as_fac[simp]:"n choose k = fac n div ((fac k) * fac (n - k))"

try

When I typed try Isabelle quickly found a counter example for me,

n = 0

k = Suc 0

Evaluated terms:

n choose k = 0

fac n div (fac k * fac (n - k)) = Suc 0

I am just wondering where  did things went wrong? Is the two calculation of
binomial coefficient not compatible with each other?

Thanks a lot.

Best,

Bob

--

Boyu Fang

```

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