Definition df-bc 14212

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30434).

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 14211	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12388	. . 3 class ℕ0
5		cz 12475	. . 3 class ℤ
6	3	cv 1540	. . . . 5 class 𝑘
7		cc0 11013	. . . . . 6 class 0
8	2	cv 1540	. . . . . 6 class 𝑛
9		cfz 13409	. . . . . 6 class ...
10	7, 8, 9	co 7352	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2113	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14182	. . . . . 6 class !
13	8, 12	cfv 6486	. . . . 5 class (!‘𝑛)
14		cmin 11351	. . . . . . . 8 class −
15	8, 6, 14	co 7352	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6486	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6486	. . . . . 6 class (!‘𝑘)
18		cmul 11018	. . . . . 6 class ·
19	16, 17, 18	co 7352	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11781	. . . . 5 class /
21	13, 19, 20	co 7352	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4474	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7354	. 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  14213