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Definition df-bc 13085
 Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 27293). 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". (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) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
Distinct variable group:   𝑘,𝑛

Detailed syntax breakdown of Definition df-bc
StepHypRef Expression
1 cbc 13084 . 2 class C
2 vn . . 3 setvar 𝑛
3 vk . . 3 setvar 𝑘
4 cn0 11289 . . 3 class 0
5 cz 11374 . . 3 class
63cv 1481 . . . . 5 class 𝑘
7 cc0 9933 . . . . . 6 class 0
82cv 1481 . . . . . 6 class 𝑛
9 cfz 12323 . . . . . 6 class ...
107, 8, 9co 6647 . . . . 5 class (0...𝑛)
116, 10wcel 1989 . . . 4 wff 𝑘 ∈ (0...𝑛)
12 cfa 13055 . . . . . 6 class !
138, 12cfv 5886 . . . . 5 class (!‘𝑛)
14 cmin 10263 . . . . . . . 8 class
158, 6, 14co 6647 . . . . . . 7 class (𝑛𝑘)
1615, 12cfv 5886 . . . . . 6 class (!‘(𝑛𝑘))
176, 12cfv 5886 . . . . . 6 class (!‘𝑘)
18 cmul 9938 . . . . . 6 class ·
1916, 17, 18co 6647 . . . . 5 class ((!‘(𝑛𝑘)) · (!‘𝑘))
20 cdiv 10681 . . . . 5 class /
2113, 19, 20co 6647 . . . 4 class ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘)))
2211, 21, 7cif 4084 . . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0)
232, 3, 4, 5, 22cmpt2 6649 . 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
241, 23wceq 1482 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  13086
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