How many different subsets, including the empty set, of the set {0, 1,

You want subsets without 0. This means you want all subsets that can be made from {1, 2, 3, 4, 5}.
Now for each of the 5 elements, you can choose to include it in your subset or not i.e. there are 2 ways of dealing with each element - include it or don't include it. So there are two choices for each of the 5 elements.
So total number of subsets will be = 2*2*2*2*2 = 32

Answer (D)

Check:
Video on Permutations: https://youtu.be/LFnLKx06EMU
Video on Combinations: https://youtu.be/tUPJhcUxllQ
Video on Probability: https://youtu.be/0BCqnD2r-kY

Is null set a subset of all the sets ?

Yes, an empty set is a subset of all sets.

Total number of subsets for a set having n elements= \(2^n\)

So total number of sets with {0, 1, 2, 3, 4, 5} = \(2^6\)
Without 0, the total number of sets for {1, 2, 3, 4, 5} = \(2^5\)

Hence the answer is \(2^6\) - \(2^5\) = \(2^5\)=32

Total number of subsets: 6 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 2^6 - 1 = 64 - 1 = 63

Number of subsets with zero = 1 + 1 x 5C1+ 1 x 5C2 + 1 x 5C3 + 1 x 5C4 = 1 + 5 + 10 + 10 + 5 = 31

Number of subsets without zero = 63 - 31 = 32

The total no of subsets that can be formed is given by2^n
Now,
Let us consider a set without 0 (1,2,3,4,5) in total there are 5 elements so possible no of sets with these 5 elements is 2^5=32
Hence
There are 32 sets which do not contain the number 0.

Please consider a kudos if this was helpful

I did not know the subset formula when trying to solve this initially - clearly it is a very simple and easy to remember formula that renders the problem easy at a mechanical level, if you understand the formula - but I am struggling to see why the alternative logic I used initially produced an incorrect answer. Can anyone explain what element is missing?

Sets consisting of 6 numbers: 0 (because one of the numbers must be excluded)
Sets consisting of 5 numbers: 1 (1,2,3,4,5)
Sets consisting of 4 numbers: 5 (combination formula; 5 choose 4)
Sets consisting of 3 numbers: 10 (combination formula; 5 choose 3)
Sets consisting of 2 numbers: 10 (combination formula; 5 chose 2)
Sets consisting of 1 number: 5 (1 or 2 or 3 or 4 or 5)

All of which sums to 31. What subset is not being picked up?

You are missing an empty set, which is a subset of all sets.

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