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Definition df-bc 13653
Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 28159).

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 ≤ 𝑘𝑛 does not hold. (Contributed by NM, 10-Jul-2005.)

Assertion
Ref Expression
df-bc C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
Distinct variable group:   𝑘,𝑛

Detailed syntax breakdown of Definition df-bc
StepHypRef Expression
1 cbc 13652 . 2 class C
2 vn . . 3 setvar 𝑛
3 vk . . 3 setvar 𝑘
4 cn0 11886 . . 3 class 0
5 cz 11970 . . 3 class
63cv 1527 . . . . 5 class 𝑘
7 cc0 10526 . . . . . 6 class 0
82cv 1527 . . . . . 6 class 𝑛
9 cfz 12882 . . . . . 6 class ...
107, 8, 9co 7145 . . . . 5 class (0...𝑛)
116, 10wcel 2105 . . . 4 wff 𝑘 ∈ (0...𝑛)
12 cfa 13623 . . . . . 6 class !
138, 12cfv 6349 . . . . 5 class (!‘𝑛)
14 cmin 10859 . . . . . . . 8 class
158, 6, 14co 7145 . . . . . . 7 class (𝑛𝑘)
1615, 12cfv 6349 . . . . . 6 class (!‘(𝑛𝑘))
176, 12cfv 6349 . . . . . 6 class (!‘𝑘)
18 cmul 10531 . . . . . 6 class ·
1916, 17, 18co 7145 . . . . 5 class ((!‘(𝑛𝑘)) · (!‘𝑘))
20 cdiv 11286 . . . . 5 class /
2113, 19, 20co 7145 . . . 4 class ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘)))
2211, 21, 7cif 4465 . . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0)
232, 3, 4, 5, 22cmpo 7147 . 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
241, 23wceq 1528 1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
Colors of variables: wff setvar class
This definition is referenced by:  bcval  13654
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