Definition df-bc 14260

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 29695).

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 14259	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12469	. . 3 class ℕ0
5		cz 12555	. . 3 class ℤ
6	3	cv 1541	. . . . 5 class 𝑘
7		cc0 11107	. . . . . 6 class 0
8	2	cv 1541	. . . . . 6 class 𝑛
9		cfz 13481	. . . . . 6 class ...
10	7, 8, 9	co 7406	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2107	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14230	. . . . . 6 class !
13	8, 12	cfv 6541	. . . . 5 class (!‘𝑛)
14		cmin 11441	. . . . . . . 8 class −
15	8, 6, 14	co 7406	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6541	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6541	. . . . . 6 class (!‘𝑘)
18		cmul 11112	. . . . . 6 class ·
19	16, 17, 18	co 7406	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11868	. . . . 5 class /
21	13, 19, 20	co 7406	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4528	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7408	. 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  14261