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The answer was given as below ,

• If we care about which way each Senator voted, then the answer is again $2^n$ : Each subcommittee defines a split + vote (those in the subcommittee vote Yes, those out vote No); and each split + vote defines de- fines a subcommittee.

• If we don’t care about which way each Senator voted, the answer is $\frac{2^n}{2} = 2^{n−1}$ .

I understood how the 1st answer can be derived. But i fail to understand the reasoning behind the second answer?

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For the second: if the 21 women in the Senate vote yes while the 79 men vote no that's the same split as the women voting no and the men yes. A very different outcome, of course.

Edit in response to comments:

Here are the four ways to split with two member committee into two blocks if the actual votes by the people in each block matter:

YES   NO
A,B
  A     B
  B     A
      A,B

Here are the two ways to split the committee if the actual votes by each block don't matter.

   A,B     empty 
     A       B
  • 0
    So 21 Women voting NO and 79 Men voting Yes is same as 40 Women voting NO and 60 Men voting Yes because it will be the same outcome – 2017-02-28
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    No. Your first bullet question asks about splits where YES and NO matter, so the splits correspond to subsets. Your second bullet question asks just about splits.If you switch the YES and NO votes you get the same SPLIT but a different OUTCOME, so there are half as many splits when you don't care which senators voted which way. – 2017-02-28
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    For eg, In a committee with 3 people, people voted as below, A- Yes , B -NO, C-Yes is the same as A - No, B-Yes, C- No ?? – 2017-02-28
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    Yes, that is the same split. There are $8$ subsets (possible sets of people who vote YES) but only $4$ splits. – 2017-02-28
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    I'm still confused. If we don't care about how each senator voted, then A- Yes,B-No,C-Yes will be same as A-No,B-Yes,C-Yes and A-Yes,B-Yes,C-No Can you help me understand with an example of 3 member committee and what are the splits will be considered finally? – 2017-02-28
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    See my edit. I just did a two member committee, which should be enough. That's the best I can do for you. I hope it helps. – 2017-02-28
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    Thanks a lot, Ethan. It helped me in understanding. In the two ways to split the committee, Can it be like `A,B empty` `B A` – 2017-02-28
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    $\{B,A\}$ is the same set as $\{A,B}$. – 2017-02-28