Definition df-bc 14228

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30414).

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 14227	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12402	. . 3 class ℕ0
5		cz 12489	. . 3 class ℤ
6	3	cv 1539	. . . . 5 class 𝑘
7		cc0 11028	. . . . . 6 class 0
8	2	cv 1539	. . . . . 6 class 𝑛
9		cfz 13428	. . . . . 6 class ...
10	7, 8, 9	co 7353	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2109	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14198	. . . . . 6 class !
13	8, 12	cfv 6486	. . . . 5 class (!‘𝑛)
14		cmin 11365	. . . . . . . 8 class −
15	8, 6, 14	co 7353	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6486	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6486	. . . . . 6 class (!‘𝑘)
18		cmul 11033	. . . . . 6 class ·
19	16, 17, 18	co 7353	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11795	. . . . 5 class /
21	13, 19, 20	co 7353	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4478	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7355	. 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  14229