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...

Topic:

Math

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`

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`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.

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