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Definition df-bc 10121
Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 11313).

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 ≤ 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 10120 . 2 class C
2 vn . . 3 setvar 𝑛
3 vk . . 3 setvar 𝑘
4 cn0 8643 . . 3 class 0
5 cz 8720 . . 3 class
63cv 1288 . . . . 5 class 𝑘
7 cc0 7329 . . . . . 6 class 0
82cv 1288 . . . . . 6 class 𝑛
9 cfz 9393 . . . . . 6 class ...
107, 8, 9co 5634 . . . . 5 class (0...𝑛)
116, 10wcel 1438 . . . 4 wff 𝑘 ∈ (0...𝑛)
12 cfa 10098 . . . . . 6 class !
138, 12cfv 5002 . . . . 5 class (!‘𝑛)
14 cmin 7632 . . . . . . . 8 class
158, 6, 14co 5634 . . . . . . 7 class (𝑛𝑘)
1615, 12cfv 5002 . . . . . 6 class (!‘(𝑛𝑘))
176, 12cfv 5002 . . . . . 6 class (!‘𝑘)
18 cmul 7334 . . . . . 6 class ·
1916, 17, 18co 5634 . . . . 5 class ((!‘(𝑛𝑘)) · (!‘𝑘))
20 cdiv 8113 . . . . 5 class /
2113, 19, 20co 5634 . . . 4 class ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘)))
2211, 21, 7cif 3389 . . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0)
232, 3, 4, 5, 22cmpt2 5636 . 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
241, 23wceq 1289 1 wff C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
Colors of variables: wff set class
This definition is referenced by:  bcval  10122
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