Question

If A = \(\{a, b, c, d, e, f\},\) then the number of subsets of A which contains at least 2 elements is 
 

• A. 64
• B. 65
• C. 57
• D. 59

Hint:

Whenever a combinatorics or probability question uses the phrase "at least", strongly consider using the complement method (Total - Unwanted). It often reduces a long series of calculations into a simple subtraction.

Answer 1

The Correct Option is C

To determine the number of subsets of set \(A = \{a, b, c, d, e, f\}\) which contain at least 2 elements, we first need to calculate the total number of subsets of \(A\). A set with \(n\) elements has \(2^n\) subsets, including the empty set and subsets with varying numbers of elements.

Here, set \(A\) has 6 elements. Therefore, the total number of subsets is given by:

\(2^6 = 64\)

This count includes:
- 1 subset of 0 elements (the empty set)
- 6 subsets of 1 element each (for each individual element in the set)
 

We need subsets containing at least 2 elements, so we subtract the subsets containing fewer than 2 elements (those subsets with 0 or 1 element).

The number of subsets with fewer than 2 elements is calculated as follows:

• 1 subset with 0 elements (empty set)
• 6 subsets with 1 element each

Total subsets with fewer than 2 elements = 1 + 6 = 7

Thus, the number of subsets with at least 2 elements is:

\(64 - 7 = 57\)

Therefore, the number of subsets of \(A\) which contain at least 2 elements is 57.