Definition df-bc 13664

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 28231).

In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". The expression (𝑁C𝐾) is read "𝑁 choose 𝐾". Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝑘 ≤ 𝑛 does not hold. (Contributed by NM, 10-Jul-2005.)

Assertion

Ref	Expression
df-bc	⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

Distinct variable group:   𝑘,𝑛

Detailed syntax breakdown of Definition df-bc

Step	Hyp	Ref	Expression
1		cbc 13663	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 11898	. . 3 class ℕ0
5		cz 11982	. . 3 class ℤ
6	3	cv 1536	. . . . 5 class 𝑘
7		cc0 10537	. . . . . 6 class 0
8	2	cv 1536	. . . . . 6 class 𝑛
9		cfz 12893	. . . . . 6 class ...
10	7, 8, 9	co 7156	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2114	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 13634	. . . . . 6 class !
13	8, 12	cfv 6355	. . . . 5 class (!‘𝑛)
14		cmin 10870	. . . . . . . 8 class −
15	8, 6, 14	co 7156	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6355	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6355	. . . . . 6 class (!‘𝑘)
18		cmul 10542	. . . . . 6 class ·
19	16, 17, 18	co 7156	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11297	. . . . 5 class /
21	13, 19, 20	co 7156	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4467	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7158	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1537	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  13665