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

Step 1: Understanding the Concept:
Given set $A = \{a, b, c, d, e, f\}$ has 6 elements.
We need to find the number of subsets containing at least 2 elements.
Step 2: Total Number of Subsets:
For a set with $n$ elements, total subsets are: \[ 2^n \] \[ 2^6 = 64 \]
Step 3: Subsets to Exclude:
We exclude subsets having less than 2 elements:

• Subsets with 0 elements: $\binom{6}{0} = 1$
• Subsets with 1 element: $\binom{6}{1} = 6$

Step 4: Required Number of Subsets:
\[ \text{Required subsets} = 64 - (1 + 6) \] \[ = 64 - 7 = 57 \]
Step 5: Final Answer:
\[ \boxed{57} \]