# Part A) only Show that for integers k and n such that `1<=k<=n` , A) `k^nC_k = n ^(n-1)C_(k-1)` B) Hence or otherwise prove that for any `x in RR` and n>=0, `sum_(k=0)^n k^nC_kx^k...

**Part A) only**

Show that for integers k and n such that `1<=k<=n` ,

A)

`k^nC_k = n ^(n-1)C_(k-1)`

B)

Hence or otherwise prove that for any `x in RR` and n>=0,

`sum_(k=0)^n k^nC_kx^k (1-x)^n-k = nx`

*print*Print*list*Cite

Expert Answers

jeew-m | Certified Educator

**`k(^nC_k) ` **

**`= k(n!)/(k!(n-k)!) ` **

**`= (n(n-1)!)/((k-1)!(n-k)!)` **

**`= n((n-1)!)/((k-1)![(n-1)-(k-1)]!)` **

**`=n(^(n-1)C_(k-1))` **

**So the answer is proved as required. **

Further Reading: