Definition df-bc 14319

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30379).

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 14318	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12499	. . 3 class ℕ0
5		cz 12586	. . 3 class ℤ
6	3	cv 1539	. . . . 5 class 𝑘
7		cc0 11127	. . . . . 6 class 0
8	2	cv 1539	. . . . . 6 class 𝑛
9		cfz 13522	. . . . . 6 class ...
10	7, 8, 9	co 7403	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2108	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14289	. . . . . 6 class !
13	8, 12	cfv 6530	. . . . 5 class (!‘𝑛)
14		cmin 11464	. . . . . . . 8 class −
15	8, 6, 14	co 7403	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6530	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6530	. . . . . 6 class (!‘𝑘)
18		cmul 11132	. . . . . 6 class ·
19	16, 17, 18	co 7403	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11892	. . . . 5 class /
21	13, 19, 20	co 7403	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4500	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7405	. 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  14320