For part (iii), why can’t I use n = 10 x 2= 20 cats?
The reason I used 20 cats instead of 10 pairs is because in part (i) my sample space was 2 cats instead of 1 pair of cats.

(i) Let X be the random variable denoting the number of cats that have blue eyes, out of 2
X~B(2,p)
P(X>=1) = 5/9
P(X=0) = 4/9
2C0(p)^0(1-p)^2 = 4/9
(1-p)^2 = 4/9
p^2 - 2p + 1 = 4/9
9p^2 - 18p + 9 = 4
9p^2 - 18p + 5 = 0 [Shown]

(iii) Let Y be the random variable denoting the number of cats in 10 cages that fulfil the event “at least one of them have blue eyes”
Y~B(20,5/9)
P(Y>=12) = 1 - P(Y<=11)
= 0.43409
= 0.434 (3s.f)

The phrasing of the Qn makes it can't be separated into individual 20 cats. You must group the cats in pairs. Coz the Qn asks "6 pairs out of 10 pairs", the condition for every pair is "at least one blue eye cat". Your working is for "at least one blue eye cat" out of 20 cats.

Thanks for the clarification