Discrete math Exercise 112 Use 12 to prove that k m n mn1 m

Discrete math

Exercise 1.12 Use (1.2) to prove that k m, (n) (m+n-1 m) n (n 1 (1.2)

Solution

P(a,b)=a!(ab)!
where a!=a×(a1)×(a2)×(a3)×....×3×21
and (ab)!=(ab)×(ab1)×(ab2)×....×2×1

So
P(n,r)=n!(nr)!
and
P(n1,r1)=(n1)!((n1)(r1))!=(n1)!(nr)!
so

nP(n1,r1)

=(n)(n1)!(nr)!

=n×(n1)×(n2)×....×2×1(nr)!

=n!(nr)!

=P(n,r)