Let,the resistance of each of the three lamp is r,
Now, lamp X and Y are in parallel combination,so their net resistance is r⋅rr+r=r2
They are again in series with lamp Z, so net resistance of the circuit is r+r2=3r2
If the voltage of the battery is V ,then current flowing through the circuit is V3r2=2V3r
So,current flowing through lamp Z is 2V3r
Now,potential drop across lamp X and Y is the same as they are in parallel combination, So the value is (V−2V3r⋅r)=V3
So,current flowing through lamp Y is V3r=V3r
So,initially, more current is flowing through lamp Z than X,so it was glowing more.
when X is no more, lamp Z and Y comes in series combination,so net resistance is r+r=2r hence,current flowing through each of them is V2r
So, this value of current is higher w.r.t initial value of current flowing through lamp Y so it becomes glower, but for Z this value has decreased than previous,so its brightness decreases.