Binomial[5,3]

Binomial[x,3]

Table[Binomial[x,n],{n,0,6}]//MatrixForm

FourD[x_]:=Binomial[x,0]+Binomial[x,1]+Binomial[x,2]+Binomial[x,3]+Binomial[x,4]

Simplify[FourD[x]]

Table[FourD[x],{x,0,6}]

SumSquare[x_]:=0Binomial[x,0]+1Binomial[x,1]+3Binomial[x,2]+2Binomial[x,3]

Simplify[SumSquare[x]]

n(n+1)(2n+1)/6

Expand[%12]

SumCubes[x_]:=0Binomial[x,0]+1Binomial[x,1]+7Binomial[x,2]+12Binomial[x,3]+6Binomial[x,4]

Simplify[SumCubes[x]]