ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-bc GIF version

Definition df-bc 10727
Description: Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 14451).

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". (๐‘C๐พ) is read "๐‘ choose ๐พ." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 โ‰ค ๐‘˜ โ‰ค ๐‘› does not hold. (Contributed by NM, 10-Jul-2005.)

Assertion
Ref Expression
df-bc C = (๐‘› โˆˆ โ„•0, ๐‘˜ โˆˆ โ„ค โ†ฆ if(๐‘˜ โˆˆ (0...๐‘›), ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))), 0))
Distinct variable group:   ๐‘˜,๐‘›

Detailed syntax breakdown of Definition df-bc
StepHypRef Expression
1 cbc 10726 . 2 class C
2 vn . . 3 setvar ๐‘›
3 vk . . 3 setvar ๐‘˜
4 cn0 9175 . . 3 class โ„•0
5 cz 9252 . . 3 class โ„ค
63cv 1352 . . . . 5 class ๐‘˜
7 cc0 7810 . . . . . 6 class 0
82cv 1352 . . . . . 6 class ๐‘›
9 cfz 10007 . . . . . 6 class ...
107, 8, 9co 5874 . . . . 5 class (0...๐‘›)
116, 10wcel 2148 . . . 4 wff ๐‘˜ โˆˆ (0...๐‘›)
12 cfa 10704 . . . . . 6 class !
138, 12cfv 5216 . . . . 5 class (!โ€˜๐‘›)
14 cmin 8127 . . . . . . . 8 class โˆ’
158, 6, 14co 5874 . . . . . . 7 class (๐‘› โˆ’ ๐‘˜)
1615, 12cfv 5216 . . . . . 6 class (!โ€˜(๐‘› โˆ’ ๐‘˜))
176, 12cfv 5216 . . . . . 6 class (!โ€˜๐‘˜)
18 cmul 7815 . . . . . 6 class ยท
1916, 17, 18co 5874 . . . . 5 class ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))
20 cdiv 8628 . . . . 5 class /
2113, 19, 20co 5874 . . . 4 class ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜)))
2211, 21, 7cif 3534 . . 3 class if(๐‘˜ โˆˆ (0...๐‘›), ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))), 0)
232, 3, 4, 5, 22cmpo 5876 . 2 class (๐‘› โˆˆ โ„•0, ๐‘˜ โˆˆ โ„ค โ†ฆ if(๐‘˜ โˆˆ (0...๐‘›), ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))), 0))
241, 23wceq 1353 1 wff C = (๐‘› โˆˆ โ„•0, ๐‘˜ โˆˆ โ„ค โ†ฆ if(๐‘˜ โˆˆ (0...๐‘›), ((!โ€˜๐‘›) / ((!โ€˜(๐‘› โˆ’ ๐‘˜)) ยท (!โ€˜๐‘˜))), 0))
Colors of variables: wff set class
This definition is referenced by:  bcval  10728
  Copyright terms: Public domain W3C validator