How many subsets of {a,b,c,d,e,f,g,h,i} include b and e?
The Correct Answer and Explanation is:
To solve this problem, we want to find how many subsets of the set S={a,b,c,d,e,f,g,h,i}S = \{a, b, c, d, e, f, g, h, i\}S={a,b,c,d,e,f,g,h,i} include both bbb and eee.
Step-by-Step Approach:
- Fix bbb and eee in the subset:
Since we are asked to find subsets that must include both bbb and eee, these two elements are always in the subset. This means we don’t have to worry about whether bbb and eee will be included in the subset or not. - Consider the remaining elements:
After including bbb and eee, we are left with the set of remaining elements {a,c,d,f,g,h,i}\{a, c, d, f, g, h, i\}{a,c,d,f,g,h,i}. This set has 7 elements. - Determine the number of subsets of the remaining set:
The number of subsets of a set with nnn elements is given by 2n2^n2n. Here, there are 7 elements remaining in the set {a,c,d,f,g,h,i}\{a, c, d, f, g, h, i\}{a,c,d,f,g,h,i}, so the number of subsets of these 7 elements is: 27=1282^7 = 12827=128 - Conclusion:
Since each of the subsets of the 7 remaining elements can combine with bbb and eee to form a subset of the original set {a,b,c,d,e,f,g,h,i}\{a, b, c, d, e, f, g, h, i\}{a,b,c,d,e,f,g,h,i}, the total number of subsets that include both bbb and eee is 128.
Final Answer:
There are 128 subsets of the set {a,b,c,d,e,f,g,h,i}\{a, b, c, d, e, f, g, h, i\}{a,b,c,d,e,f,g,h,i} that include both bbb and eee.
