Question

If A = {1, 2, 3, 4, 5, 6}, then the number of subsets of A which contain atleast two elements is

• A. 64
• B. 63
• C. 57
• D. 58

Answer 1

The Correct Option is C

If \(A = \{1, 2, 3, 4, 5, 6\}\), then the number of subsets of \(A\) which contain at least two elements is:

The total number of subsets of A is \(2^6 = 64\). The number of subsets with at least two elements is equal to the total number of subsets minus the number of subsets with zero elements (the empty set) and the number of subsets with one element.

Number of subsets with zero elements = 1 (the empty set)

Number of subsets with one element = 6 (one for each element)

Number of subsets with at least two elements = \(64 - 1 - 6 = 57\)

Answer 2

The total number of subsets of $ A $ is:

$$ 2^{|A|} = 2^6 = 64. $$

The number of subsets with 0 elements is:

$$ \binom{6}{0} = 1 \quad \text{(the empty set)}. $$

The number of subsets with 1 element is:

$$ \binom{6}{1} = 6. $$

Thus, the number of subsets with at least two elements is:

$$ 64 - 1 - 6 = 57. $$