D.2.2.4 binomial

Procedure from library general.lib (see general_lib).

Usage:

binomial(n,k[,p]); n,k,p integers

Return:

binomial(n,k); binomial coefficient n choose k
- of type string (computed in characteristic 0)
binomial(n,k,p); n choose k, computed in characteristic 0 or prime(p)
- of type number if a basering, say R, is present and p=0=char(R) or if prime(p)=char(R)
- of type string else

Note:

In any characteristic, binomial(n,k) = coefficient of x^k in (1+x)^n

Example:

LIB "general.lib";
binomial(200,100);"";                   //type string, computed in char 0
→ 90548514656103281165404177077484163874504589675413336841320
→ 
binomial(200,100,3);"";                 //type string, computed in char 3
→ 0
→ 
int n,k = 200,100;
ring r = 0,x,dp;
number b1 = binomial(n,k,0);            //type number, computed in ring r
poly b2 = coeffs((x+1)^n,x)[k+1,1];     //coefficient of x^k in (x+1)^n
b1-b2;                                  //b1 and b2 should coincide
→ 0

See also: prime.