Question

A is a set containing n elements. A subset P₁ is chosen, and A is reconstructed by replacing the elements of P₁. The same process is repeated for subsets P₁, P₂, … , P_m, with m > 1. The Number of ways of choosing P₁, P₂, …, P_m so that P₁Č P₂Č … Č P_m= A is –

• A. (2^m – 1)^mn
• B. ^m+nC_m
• C. (2ⁿ – 1)^m
• D. None

Answer 1

Answer: (D)

None

Answer 2

Let A = {a₁, a₂,…..a_n}

For each a_i (1 £ i £ n), either a_i Ī P_j or a_i Ļ P_j (1 £ j £ m) . Thus, there are 2^m choices in which a_i (1 £ j £ n) may belong to the P_j ¢s.

Also there is exactly one choice, viz., a_i Ļ P_j for j = 1, 2, …, m, for which a_i Ļ P₁ Č P₂ Č...Č P_m.

Therefore, a_i Ī P₁ Č P₂ Č …. Č P_m in (2^m – 1) ways . Since there are n elements in the set A, the number of ways of constructing subsets

P₁, P₂, ….. , P_m is (2^m – 1)ⁿ