Definition df-bc 14359

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30876).

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 14358	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12521	. . 3 class ℕ0
5		cz 12608	. . 3 class ℤ
6	3	cv 1569	. . . . 5 class 𝑘
7		cc0 11117	. . . . . 6 class 0
8	2	cv 1569	. . . . . 6 class 𝑛
9		cfz 13553	. . . . . 6 class ...
10	7, 8, 9	co 7419	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2146	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14329	. . . . . 6 class !
13	8, 12	cfv 6540	. . . . 5 class (!‘𝑛)
14		cmin 11458	. . . . . . . 8 class −
15	8, 6, 14	co 7419	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6540	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6540	. . . . . 6 class (!‘𝑘)
18		cmul 11122	. . . . . 6 class ·
19	16, 17, 18	co 7419	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11888	. . . . 5 class /
21	13, 19, 20	co 7419	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4489	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7421	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1570	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is used by:  bcval  14360