Question

If \(n(A) = 8\), then the number of subsets of \(A\) which contain \(2\) or \(6\) elements is

• A. \(24\)
• B. \(28\)
• C. \(48\)
• D. \(56\)
• E. \(216\)

Hint:

For a set with \(n\) elements, the number of subsets with exactly \(r\) elements is \({}^{n}C_{r}\). Also remember the useful identity \({}^{n}C_{r} = {}^{n}C_{n-r}\).

Answer 1

The Correct Option is D

Step 1: Understand what \(n(A) = 8\) means.
The notation \(n(A) = 8\) means that the set \(A\) has \(8\) elements. We need to count subsets having exactly \(2\) elements or exactly \(6\) elements.

Step 2: Recall the formula for choosing subsets of fixed size.
The number of subsets of an \(8\)-element set containing exactly \(r\) elements is:
\[ {}^{8}C_{r} \] So here we need:
\[ {}^{8}C_{2} + {}^{8}C_{6} \]

Step 3: Calculate the number of 2-element subsets.
\[ {}^{8}C_{2} = \frac{8!}{2!6!} = \frac{8 \cdot 7}{2 \cdot 1} = 28 \]

Step 4: Calculate the number of 6-element subsets.
\[ {}^{8}C_{6} = \frac{8!}{6!2!} = \frac{8 \cdot 7}{2 \cdot 1} = 28 \] Also, by symmetry of combinations, \({}^{8}C_{6} = {}^{8}C_{2}\).

Step 5: Add both counts.
Now total number of required subsets is:
\[ {}^{8}C_{2} + {}^{8}C_{6} = 28 + 28 = 56 \]

Step 6: Interpret the result.
So, there are \(56\) subsets of \(A\) that contain either \(2\) elements or \(6\) elements.

Step 7: Match with the options.
The value \(56\) is given in option (4). Therefore, the correct answer is option (4).