Definition df-bc 14370

Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30935).

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 14369	. 2 class C
2		vn	. . 3 setvar 𝑛
3		vk	. . 3 setvar 𝑘
4		cn0 12531	. . 3 class ℕ0
5		cz 12618	. . 3 class ℤ
6	3	cv 1569	. . . . 5 class 𝑘
7		cc0 11127	. . . . . 6 class 0
8	2	cv 1569	. . . . . 6 class 𝑛
9		cfz 13564	. . . . . 6 class ...
10	7, 8, 9	co 7414	. . . . 5 class (0...𝑛)
11	6, 10	wcel 2145	. . . 4 wff 𝑘 ∈ (0...𝑛)
12		cfa 14340	. . . . . 6 class !
13	8, 12	cfv 6533	. . . . 5 class (!‘𝑛)
14		cmin 11468	. . . . . . . 8 class −
15	8, 6, 14	co 7414	. . . . . . 7 class (𝑛 − 𝑘)
16	15, 12	cfv 6533	. . . . . 6 class (!‘(𝑛 − 𝑘))
17	6, 12	cfv 6533	. . . . . 6 class (!‘𝑘)
18		cmul 11132	. . . . . 6 class ·
19	16, 17, 18	co 7414	. . . . 5 class ((!‘(𝑛 − 𝑘)) · (!‘𝑘))
20		cdiv 11898	. . . . 5 class /
21	13, 19, 20	co 7414	. . . 4 class ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘)))
22	11, 21, 7	cif 4482	. . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)
23	2, 3, 4, 5, 22	cmpo 7416	. 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
24	1, 23	wceq 1570	1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))

This definition is used by:  bcval  14371