Pamatujte: Suma (∑_(i=Start)^(i=EndGlobal1))body[i,f(iEndGlobal2)]
je For cyklus 
sum=0;
for i:=Start to EndGlobal1
do
begin
hratky s i;
sum:=sum+body[i,iEndGlobal2];
end;

pokud transformujeme i, tak se nesmi zmenit pocet

rule:=(({{n_Integer}, {k_Integer}})):>Binomial[n,k];
FullSimplify[((n!)/(k!*(n-k)!))-Binomial[n,k]]

{(({{7}, {3}}))/.rule,Binomial[7,3]}

n=5;
lhs=(∑_(k=0)^(n+1))((({{n+1}, {k}}))/.rule)(x^k)
rhs1=(∑_(k=0)^n)((({{n}, {k}}))/.rule)(x^k)+(∑_(k=0)^n)((({{n}, {k}}))/.rule)(x^(k+1))

(∑_(k=0)^n)((({{n}, {k}}))/.rule)*(x^(k+1))
(∑_(j=1)^(n+1))((({{n}, {j-1}}))/.rule)*(x^j)
(∑_(k=1)^(n+1))((({{n}, {k-1}}))/.rule)*(x^k)

rhsSpatne=(∑_(k=0)^n)((({{n}, {k}}))/.rule)(x^k)+(∑_(k=1)^(n+1))((({{n+1}, {k-1}}))/.rule)*(x^k)
(*n v mezi sumace se transformuje jinak, nez n v argumetu sumace!!!!!!*)

rhs2=(∑_(k=0)^n)((({{n}, {k}}))/.rule)(x^k)+(∑_(k=1)^(n+1))((({{n}, {k-1}}))/.rule)*(x^k)

(∑_(k=0)^(n+1))((({{n+1}, {k}}))/.rule)(x^k)-(((({{n+1}, {n+1}}))/.rule)*(x^(n+1))+((({{n+1}, {0}}))/.rule)*(x^0)+(∑_(k=1)^n)((({{n+1}, {k}}))/.rule)(x^k))

((({{n+1}, {n+1}}))/.rule)*(x^(n+1))+((({{n+1}, {0}}))/.rule)*(x^0)

(∑_(k=0)^n)((({{n}, {k}}))/.rule)(x^k)-(((({{n}, {0}}))/.rule)*(x^0)+(∑_(k=1)^n)((({{n}, {k}}))/.rule)(x^k))

(∑_(k=1)^(n+1))((({{n}, {k-1}}))/.rule)(x^k)-(((({{n}, {n}}))/.rule)*(x^(n+1))+(∑_(k=1)^n)((({{n}, {k-1}}))/.rule)(x^k))

(∑_(k=1)^n)((({{n+1}, {k}}))/.rule)(x^k)-((∑_(k=1)^n)((({{n}, {k}}))/.rule)(x^k)+(∑_(k=1)^n)((({{n}, {k-1}}))/.rule)(x^k))

(*ClearAll[n];
FullSimplify[Binomial[n+1,k]-(Binomial[n,k-1]+Binomial[n,k])]*)