Definition df-bc 14265

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30522).

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 14264	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12437	. . 3 class ℕ0
5		cz 12524	. . 3 class ℤ
6	3	cv 1541	. . . . 5 class 𝑘
7		cc0 11038	. . . . . 6 class 0
8	2	cv 1541	. . . . . 6 class 𝑛
9		cfz 13461	. . . . . 6 class ...
10	7, 8, 9	co 7367	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2114	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14235	. . . . . 6 class !
13	8, 12	cfv 6498	. . . . 5 class (!‘𝑛)
14		cmin 11377	. . . . . . . 8 class −
15	8, 6, 14	co 7367	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6498	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6498	. . . . . 6 class (!‘𝑘)
18		cmul 11043	. . . . . 6 class ·
19	16, 17, 18	co 7367	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11807	. . . . 5 class /
21	13, 19, 20	co 7367	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4466	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7369	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1542	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  14266