**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`

### 1 Answer | Add Yours

**`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. **

**Sources:**

### Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes