Explanation
Solving:
1. Given Set:
A={1,2,3,4,5,6,7}
Total elements n(A)=7
2. Condition:
Subsets must contain both 1 and 7.
Fixed elements = {1,7}
Remaining elements to choose from = {2,3,4,5,6}
Number of remaining elements = 5
3. Total Subsets containing 1 and 7:
Total subsets =25=32
4. Proper Subsets:
A proper subset cannot be the set itself.
Since the original set A contains both 1 and 7, it is included in the 32 subsets calculated above.
To get proper subsets, we subtract this one case:
Proper subsets =32−1=31
Correct Option:
(b) 31
