If Pr n = Pr + 1 n and Cr n = Cr - 1 n , then the value of r is...

We have, $Pr n = Pr + 1 n$

$⇒ \dfrac{n !}{\left(n - r\right) !} = \dfrac{n !}{\left(n - r - 1\right) !}$

$⇒ \left(n - r\right) = 1 \ldots \left(1\right)$

and $Cr n = Cr - 1 n$

$⇒ \dfrac{n !}{r ! \left(n - r\right) !} = \dfrac{n !}{\left(r - 1\right) ! \left(n - r + 1\right) !}$

$⇒ \dfrac{1}{r\left(n - r\right) !} = \dfrac{1}{\left(n - r + 1\right)\left(n - r\right) !}$

$⇒ n - r + 1 = r$

$⇒ n + 1 = 2r \ldots \left(2\right)$

From equation \left(1\right) \text{$and$} \left(2\right),$we get</p><p>$2r - 1 - r = 1$</p><p>$⇒ r = 2 .