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Definition df-bc 13130
Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 27439).

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 13129 . 2 class C
2 vn . . 3 setvar 𝑛
3 vk . . 3 setvar 𝑘
4 cn0 11330 . . 3 class 0
5 cz 11415 . . 3 class
63cv 1522 . . . . 5 class 𝑘
7 cc0 9974 . . . . . 6 class 0
82cv 1522 . . . . . 6 class 𝑛
9 cfz 12364 . . . . . 6 class ...
107, 8, 9co 6690 . . . . 5 class (0...𝑛)
116, 10wcel 2030 . . . 4 wff 𝑘 ∈ (0...𝑛)
12 cfa 13100 . . . . . 6 class !
138, 12cfv 5926 . . . . 5 class (!‘𝑛)
14 cmin 10304 . . . . . . . 8 class
158, 6, 14co 6690 . . . . . . 7 class (𝑛𝑘)
1615, 12cfv 5926 . . . . . 6 class (!‘(𝑛𝑘))
176, 12cfv 5926 . . . . . 6 class (!‘𝑘)
18 cmul 9979 . . . . . 6 class ·
1916, 17, 18co 6690 . . . . 5 class ((!‘(𝑛𝑘)) · (!‘𝑘))
20 cdiv 10722 . . . . 5 class /
2113, 19, 20co 6690 . . . 4 class ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘)))
2211, 21, 7cif 4119 . . 3 class if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0)
232, 3, 4, 5, 22cmpt2 6692 . 2 class (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
241, 23wceq 1523 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  13131
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