If n is the number of ways five different employees can sit into...

Given,

Rooms are identical,

So, total ways to distribute $5$ employees into $4$ rooms are:

$5, 0, 0, 0 ⇒ 1$ way

$4, 1, 0, 0 ⇒ \dfrac{5 !}{4 !} = 5$ ways

$3, 2, 0, 0 ⇒ \dfrac{5 !}{3 ! 2 !} = 10$ ways

$2, 2, 0, 1 ⇒ \dfrac{5 !}{2 ! 2 ! 2 !} = 15$ ways

$2, 1, 1, 1 ⇒ \dfrac{5 !}{2 ! {\left(1 !\right)}^{3}3 !} = 10$ ways

$3, 1, 1, 0 ⇒ \dfrac{5 !}{3 ! 2 !} = 10$ ways

Total $⇒ 1 + 5 + 10 + 15 + 10 + 10 = 51$ ways