n!/{i!(n-i)!} = C[n,i] と表せば
iC[n,i] = nC[n-1,i-1]

(i=0 -> n) i* (p^i)/(i!) * ((1-p)^(n-i)) / ((n-i)!)
= (1/n!)(i=0 -> n) i*C[n,i] (p^i)* ((1-p)^(n-i))
= (1/n!)(i=1 -> n) n*C[n-1,i-1] (p^i)* ((1-p)^(n-i))
= (1/n!)(i=0 -> n-1) np*C[n-1,i] (p^i)* ((1-p)^(n-1-i))
= (np/n!)
= p/(n-1)!