Definition df-bc 14259

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30540).

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 14258	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12431	. . 3 class ℕ0
5		cz 12518	. . 3 class ℤ
6	3	cv 1541	. . . . 5 class 𝑘
7		cc0 11032	. . . . . 6 class 0
8	2	cv 1541	. . . . . 6 class 𝑛
9		cfz 13455	. . . . . 6 class ...
10	7, 8, 9	co 7361	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2114	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14229	. . . . . 6 class !
13	8, 12	cfv 6493	. . . . 5 class (!‘𝑛)
14		cmin 11371	. . . . . . . 8 class −
15	8, 6, 14	co 7361	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6493	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6493	. . . . . 6 class (!‘𝑘)
18		cmul 11037	. . . . . 6 class ·
19	16, 17, 18	co 7361	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11801	. . . . 5 class /
21	13, 19, 20	co 7361	. . . 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 7363	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1542	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  14260