: nPr = n-1Pr+r.n-1Pr-1
You asked:
Given \(n P r = n - 1 P r + r\) evaluate the expression \(n - 1 P r - 1\).
MathBot Answer:
\[- 1 P r + n - 1 = n - 1 - \frac{n + r}{n + 1}\]
\[P n r = - P r + n + r\]Solve for $P$: \[P = \frac{n + r}{r \left(n + 1\right)}\]Substitute $P$ into the given expression: $$\begin{aligned}- 1 P r + n - 1 &= n - r \frac{n + r}{r \left(n + 1\right)} - 1 \\ - 1 P r + n - 1 &= n - 1 - \frac{n + r}{n + 1}\end{aligned}$$