Definition df-bc 13382

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 27866).

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 13381	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 11617	. . 3 class ℕ0
5		cz 11703	. . 3 class ℤ
6	3	cv 1657	. . . . 5 class 𝑘
7		cc0 10251	. . . . . 6 class 0
8	2	cv 1657	. . . . . 6 class 𝑛
9		cfz 12618	. . . . . 6 class ...
10	7, 8, 9	co 6904	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2166	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 13352	. . . . . 6 class !
13	8, 12	cfv 6122	. . . . 5 class (!‘𝑛)
14		cmin 10584	. . . . . . . 8 class −
15	8, 6, 14	co 6904	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6122	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6122	. . . . . 6 class (!‘𝑘)
18		cmul 10256	. . . . . 6 class ·
19	16, 17, 18	co 6904	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11008	. . . . 5 class /
21	13, 19, 20	co 6904	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4305	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpt2 6906	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1658	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is referenced by:  bcval  13383