(*The binomial coefficients.*)

Binomial[8,4]

TableForm@Table[Binomial[n,k], {n,0,12,1},{k,0,n,1}]

Manipulate[{n,k,Binomial[n,k]}, {n,0,40,1},{k,0,40,1}]

(***********************************************)

(* Here is the solution to Lecture 4, Problem 1.4. We let n be the number of Independents.*)

(*
The number of ways of selecting a sample of size 100 is Binomial[4n, 100].  The number of ways that the sample could contain 25 Independents and 75 party-members is Binomial[n,25]Binomial[3n, 75].  Thus, the probability of the sample containing 25 Independents is the following:
*)

P[n_]:=Binomial[n,25]Binomial[3n, 75]/Binomial[4n, 100]

(*If there 25 Independents, then there are 100 voters and there is only one sample of size 100.*)

N[P[25]]

(*If there 26 Independents, there are 104 voters.  Now the probability of having 25 Independents in the sample is much less.*)

N[f[26]]

N[f[100]]

N[f[1000]]

N[f[10000]]

N[f[1000000]]