Show that (n+1) ^n P_r=(n-r+1)^ (n+1) P_r. - Vidyadip

Question

Show that $(n+1){ }^n P_r=(n-r+1)^{(n+1)} P_r$.

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Answer

$
\begin{aligned}
& \text { L.H.S. }=(n+1){ }^n P_r=(n+1) \frac{n !}{(n-r) !}=\frac{(n+1) !}{(n-r) !} \\
& \begin{aligned}
& \text { R.H.S. }=(n-r+1){ }^{(n+1)} P_r=(n-r+1) \frac{(n+1) !}{(n-r+1) !} \\
& \\
&=\frac{(n-r+1)(n+1) !}{(n-r+1)(n-r) !} \\
&=\frac{(n+1) !}{(n-r) !}
\end{aligned} \\
& \therefore \quad \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$

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