Definition df-bc 14268

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30381).

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 14267	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12442	. . 3 class ℕ0
5		cz 12529	. . 3 class ℤ
6	3	cv 1539	. . . . 5 class 𝑘
7		cc0 11068	. . . . . 6 class 0
8	2	cv 1539	. . . . . 6 class 𝑛
9		cfz 13468	. . . . . 6 class ...
10	7, 8, 9	co 7387	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2109	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14238	. . . . . 6 class !
13	8, 12	cfv 6511	. . . . 5 class (!‘𝑛)
14		cmin 11405	. . . . . . . 8 class −
15	8, 6, 14	co 7387	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6511	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6511	. . . . . 6 class (!‘𝑘)
18		cmul 11073	. . . . . 6 class ·
19	16, 17, 18	co 7387	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11835	. . . . 5 class /
21	13, 19, 20	co 7387	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4488	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7389	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1540	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  14269