@pratik0809 It is asked to create subset so that NO two elements sum to 11. So Null set satisfies right ?
Logic here is, for a usual case (without any conditions), we can have 2^n subsets for a set of n elements. Why ? because each element can either be present or not in the subset. So each element can have two options giving a total of 2^n possibilities.
Here, we have 5 pairs and each pair can contribute in 3 ways - First element selected or Second element selected or None is selected.
So total subsets = 3^5 = 243 ways (which includes null set too)

p = prob (overnight born = boy| Next day chosen = boy)
= prob (overnight born = boy AND Next day chosen = boy) / prob (Next day chosen = boy)
= .5 * 4/T / (.5*4/T + .5 * 3/T) = 2/3.5 = 4/7 ~ .57

Each angle is 180(p-2)/p.
180-{360}/{p} = k
So 360/p has to be an integer.
360 = 2^3 * 3^2 * 5^1
So there are 4 * 3 * 2 = 24 possibilities, but we exclude 1 and 2, because p > = 3
So , 24 -2 = 22
Hence, choice (c) is the right answer

The roots are a and b:
a + b = p and ab = 12
(a + b)^2 = p^2
(a - b)^2 = (a + b)^2 - 4ab
=> (a - b)^2 = p^2 - 12 * 4 = p^2 - 48
If |a - b| ≥ 12 { Difference between the roots is at least 12}
then, (a - b)^2 ≥ 144
p^2 - 48 ≥ 144
p^2 ≥ 192
P ≥ 8√3 or P ≤ -8√3