A farmer has some hens and some rabbits. If the total number of the feet of the hens and the rabbits is $160$ and their total number of heads is $50$, the number of hens the farmer has is:

Let the number of hens be $H$ and the number of rabbits be $R$.
Each hen has $2$ feet and $1$ head. Each rabbit has $4$ feet and $1$ head.

From heads condition:

$$H+R=50⟹R=50-H\text{— (Equation 1)}$$

From feet condition:

$$2H+4R=160\text{— (Equation 2)}$$

Substitute Equation 1 into Equation 2:

$$2H+4(50-H)=160$$

$$2H+200-4H=160$$

$$-2H=160-200=-40$$

$$2H=40⟹H=20$$

Thus, the farmer has $20$ hens.