Definition df-bc 14226

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30527).

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 14225	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12401	. . 3 class ℕ0
5		cz 12488	. . 3 class ℤ
6	3	cv 1540	. . . . 5 class 𝑘
7		cc0 11026	. . . . . 6 class 0
8	2	cv 1540	. . . . . 6 class 𝑛
9		cfz 13423	. . . . . 6 class ...
10	7, 8, 9	co 7358	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2113	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14196	. . . . . 6 class !
13	8, 12	cfv 6492	. . . . 5 class (!‘𝑛)
14		cmin 11364	. . . . . . . 8 class −
15	8, 6, 14	co 7358	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6492	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6492	. . . . . 6 class (!‘𝑘)
18		cmul 11031	. . . . . 6 class ·
19	16, 17, 18	co 7358	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11794	. . . . 5 class /
21	13, 19, 20	co 7358	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4479	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7360	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1541	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  14227