Solve the following

Question:

The value of n – 1Pr + r  n – 1P– 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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