**Answer:** 10 cockroaches, 5 mice and 35 birds.

Cockroaches have 6 feet, mice have 4 and birds have 2.

For every mouse there are two cockroaches, so every mouse is worth 3 heads and 16 feet (4 + 6 + 6).

We can now write down an expression for the heads and the feet (calling birds B and mice M):

Heads gives us:

B + 3M = 50 (1)

Feet gives us:

2B + 16M = 150 (2)

If we double (1) we get:

2B + 6M = 100 (3)

We can now do (2) - (3) to give:

10M = 50

M = 5

So we have 5 mice (and 10 cockroaches). We can use M = 5 in (1) to give:

B + 3 x 5 = 50

B + 15 = 50

B = 35

So, C = 10, M = 5 and B = 35. Checking that 10 + 5 + 35 = 50, and 10 x 6 + 5 x 4 + 35 x 2 = 150. QED.