Question:
The value of n – 1Pr + r n – 1Pr – 1 is __________.
Solution:
${ }^{n-1} P_{r}+r^{n-1} P_{r-1}$
$=\frac{(n-1) !}{(n-1-r) !}+r \frac{(n-1) !}{(n-1-r+1) !}$
$=\frac{(n-1) !}{(n-r-1) !}+r \frac{(n-1) !}{(n-r) !}$
$=(n-1) !\left[\frac{1}{(n-r-1) !}+\frac{r}{(n-r)(n-r-1) !}\right]$
$=(n-1) !\left[\frac{n-r+r}{(n-r)(n-r-1) !}\right]$
$=\frac{(n-1) !(n-r+r)}{(n-r) !}=\frac{(n-1) ! n}{(n-r) !}={ }^{n} P_{r}$
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