Definition df-bc 14207

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30427).

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 14206	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12378	. . 3 class ℕ0
5		cz 12465	. . 3 class ℤ
6	3	cv 1540	. . . . 5 class 𝑘
7		cc0 11003	. . . . . 6 class 0
8	2	cv 1540	. . . . . 6 class 𝑛
9		cfz 13404	. . . . . 6 class ...
10	7, 8, 9	co 7346	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2111	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14177	. . . . . 6 class !
13	8, 12	cfv 6481	. . . . 5 class (!‘𝑛)
14		cmin 11341	. . . . . . . 8 class −
15	8, 6, 14	co 7346	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6481	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6481	. . . . . 6 class (!‘𝑘)
18		cmul 11008	. . . . . 6 class ·
19	16, 17, 18	co 7346	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11771	. . . . 5 class /
21	13, 19, 20	co 7346	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4475	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7348	. 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  14208