Question

Let T contains 93 elements. A, B are subsets of T such that
(i) A ≠ B
(ii) A ∪ B = T
How many unordered pairs (A, B) will exist. ( (A,B) & (B,A) are same)

विकल्प

• `2^93`

• `3^93 - 1`

• `2^92`

• None of these

Answer 1

Let A contains x, (0 ≤ x ≤ 93) elements: so, A can be chosen in ⁹³C_x ways.
For each such A, the set B must necessarily have remaining (93 - x) elements & possible some elements of A. for every element of A to be in B,
we have 2 choices: either we can take or we can leave. So, B can be chosen in 2x ways.
So, there are ⁹³C_x 2^x ways to choose A & B where 0 ≤ x ≤ 93.

⇒ `sum_(x = 0)^93 ""^93C_x 2^x = ( 1 +2 )^93`

= 3⁹³
There is 1 case where A = B = (T). & since order does not matter so, required answer

= `(3^93 - 1)/2`

None of these is the correct option.